Quantised Vortices: A Handbook of Topological Excitations [Concise ed.] 1643271237, 9781643271231

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Quantised Vortices A handbook of topological excitations

Quantised Vortices A handbook of topological excitations Tapio Simula Swinburne University of Technology, Melbourne, Australia

Morgan & Claypool Publishers

Copyright ª 2019 Morgan & Claypool Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organizations. Rights & Permissions To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact [email protected]. ISBN ISBN ISBN

978-1-64327-126-2 (ebook) 978-1-64327-123-1 (print) 978-1-64327-124-8 (mobi)

DOI 10.1088/2053-2571/aafb9d Version: 20190701 IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print) A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA, 94901, USA IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

Dedication Ollille

Contents Preface

xiii

Acknowledgements

xvi

Author biography

xvii

Part I

Vortices in Flatland

1

Vortices

1-1

1.1 1.2 1.3 1.4

Space–time symmetries Quantum liquids Vorticity in classical fluids Vorticity in quantum liquids

1-1 1-2 1-3 1-4

2

Quasiparticle picture

2-1

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21

Emergence of quasiparticles Boson commutation relations Fermion anticommutation relations Majorana relations Anyon quasiparticles Non-abelian anyon quasiparticles Bogoliubov–de Gennes equations Time-reversal symmetry Particle–hole symmetry Chiral symmetry Phonon spectrum Landau critical velocity Roton–maxon spectrum Edge modes Dipole, breathing, quadrupole and scissors modes Kelvin mode vortex waves Tkachenko mode vortex waves Caroli–de Gennes–Matricon modes Nambu–Goldstone zero mode Majorana zero mode Magnon spin waves

vii

2-1 2-2 2-2 2-3 2-3 2-4 2-4 2-5 2-6 2-7 2-8 2-9 2-9 2-10 2-11 2-12 2-14 2-15 2-15 2-16 2-17

Quantised Vortices

3

Cold atoms

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

Scalar Bose–Einstein condensates Bose zero-temperature energy functional Thomas–Fermi relations Healing length Thermodynamic relations Quantum hydrodynamic equations Two-component Bose–Einstein condensates Spin-1 Bose–Einstein condensates Spin-2 Bose–Einstein condensates High-spin Bose–Einstein condensates Representations of spinor Bose–Einstein condensates Exotic interactions Bardeen–Cooper–Schrieffer mean-field theory Ultracold Fermi gases Dirac–Bogoliubov–de Gennes systems Gapless, massless, linear spectra Gapped, massive, quadratic spectra

4

Topological invariants and quantities

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

Topology and ordered structures A game of lines and loops Maps and order parameters Homotopy classification of defects Burgers vector Gauss–Bonnet theorem Winding number Berry phase, curvature, and connection Chern number Linking number, writhe and twist Helicity Enstrophy Kauffman bracket polynomial Jones polynomial

3-1

viii

3-1 3-3 3-4 3-4 3-5 3-6 3-7 3-8 3-10 3-11 3-12 3-13 3-14 3-15 3-17 3-17 3-18 4-1 4-1 4-1 4-2 4-3 4-4 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-10 4-10

Quantised Vortices

5

Topological excitations

5-1

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13

Topological defects Soliton Bright soliton Grey and dark soliton Solitonic vortex Plain vortex Polynomial vortex Coherence vortex Fractional vortex Baby skyrmion Monopole Fluxon, chargeon, and dyon Alice vortex and Cheshire charge

5-1 5-1 5-2 5-2 5-3 5-4 5-5 5-6 5-6 5-7 5-8 5-8 5-9

6

Structure of a plain vortex

6-1

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

Vortex uncertainty principle Kelvon Circulation quantum Vortex energy Thermodynamic stability Spectral, energetic stability Dynamical Lyapunov stability Inertial vortex mass Gravitational vortex mass Kelvon-based vortex mass Hydrodynamic induced vortex mass component Relativistic vortex mass component Baym–Chandler vortex mass Kopnin vortex mass

6-1 6-2 6-3 6-3 6-4 6-5 6-5 6-6 6-7 6-7 6-8 6-8 6-9 6-9

7

Vortex dynamics

7-1

7.1 7.2 7.3 7.4 7.5

Adiabatic vortex dynamics Vortex force and velocity Magnus effect and mutual induction Vortex pair creation and annihilation Onsager point vortex model

7-1 7-2 7-3 7-3 7-4

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Quantised Vortices

7-5 7-6 7-7 7-8 7-9 7-9 7-10

7.6 7.7 7.8 7.9 7.10 7.11 7.12

Vortex–particle duality Point vortex model with cylindrical boundary Point vortex models with square boundaries Point vortex models in general domains Vortex classification algorithm Vortex temperature Winding number scaling laws

8

Vortex production in Bose–Einstein condensates

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19

Coherent coupling of internal states Laguerre–Gauss laser modes Topological angular momentum conversion Rotating bucket Rotating thermal cloud Stirring Shaking bucket Snaking instability Many-wave interference Vortex–antivortex honeycomb lattices Caustics and diffraction catastrophes Vortex quasicrystals Vortex phasons Vortex Moiré superlattices Synthetic gauge fields Optical flux lattices Filtered speckle fields Kibble–Zurek mechanism and quenches Berezinskii–Kosterlitz–Thouless mechanism

9

Topological quantum computation

9-1

9.1 9.2 9.3 9.4 9.5 9.6

Non-abelian anyons Topological qubits Quantum dimension Majorana Ising anyon model Fibonacci anyon model Model k anyons

9-1 9-2 9-2 9-4 9-4 9-5

x

8-1 8-1 8-2 8-3 8-3 8-4 8-5 8-5 8-5 8-6 8-6 8-8 8-9 8-9 8-10 8-11 8-12 8-13 8-14 8-15

Quantised Vortices

9.7 9.8 9.9 9.10 9.11 9.12

Non-abelian vortex anyons Annihilation, pass-through and rungihilation Non-abelian vortex anyon models Vortex anyon creation, pinning, braiding, and fusion From quantum circuits to anyon braiding Evaluation of space–time knot invariants

9-6 9-7 9-8 9-8 9-9 9-10

10

Two-dimensional quantum turbulence

10-1

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21

11 11.1 11.2 11.3 11.4 11.5

Regular and chaotic few-vortex dynamics Inverse energy and direct enstrophy cascades Vortex near-field spectrum Vortex far-field spectrum Vortex dipole spectrum Kolmogorov–Obukhov spectrum Onsager vortex spectrum Spin turbulence spectrum Helmholtz decomposition Enstrophy conservation and non-conservation Evaporative heating of vortices Point vortex model of turbulence Non-abelian two-dimensional quantum turbulence Superfluid Reynolds number Eddy turnover time Anomalous hydrodynamics of vortices Negative absolute temperature Negative absolute vortex temperature Non-thermal fixed point Dynamical phase transitions Condensation of Onsager vortices

Vortex states of matter in Flatland

10-1 10-2 10-3 10-3 10-4 10-4 10-4 10-5 10-6 10-6 10-7 10-8 10-8 10-9 10-10 10-10 10-11 10-11 10-12 10-12 10-13 11-1 11-1 11-2 11-2 11-3 11-3

BCS superconductivity Meissner effect Type-II superconductors Abrikosov vortex lattice Vortex pinning and creep motion

xi

Quantised Vortices

11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14

Vortex matter in rotating superfluids Vortex nucleation and Hess–Fairbank effect Vortex lattices in neutral superfluids Feynman rule Vortex lattice melting Two-dimensional vortex Coulomb gas Two-dimensional Coulomb gas: quantum Hall effects Two-dimensional Coulomb gas: Hauge–Hemmer transition Two-dimensional Coulomb gas: Berezinskii–Kosterlitz–Thouless transition 11.15 Two-dimensional Coulomb gas: supercondensation transition 11.16 Two-dimensional Coulomb gas: Einstein–Bose condensation transition

12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15

Superfluid universe

11-4 11-5 11-5 11-6 11-6 11-7 11-8 11-9 11-10 11-11 11-12

12-1

Vacuum Speed of light Photon Particles and antiparticles Positronium Pair creation and annihilation Photon emission and absorption Charge Spin Dipole moment Electrodynamics Non-abelian fractional charge particles Quantum chromodynamics Gravitation and black holes Cosmic inflation

xii

12-2 12-2 12-2 12-3 12-3 12-3 12-4 12-4 12-4 12-5 12-5 12-6 12-6 12-7 12-7

Preface This book is a subjective reflection of the authorʼs own research interests over the past couple of decades. The subject matter is divided into two parts. Part I: Vortices in Flatland (this book) mainly focuses on the physics of vortices in two-dimensional space. Part II: Vortices in Spaceland (in preparation), will mainly focus on the physics of vortices in three-dimensional space. The emphasis in both parts is on vortices in Bose–Einstein condensates (BECs) described by a complex valued scalar order parameter. The generic topological concepts pertinent to such systems and other quantum liquids are applicable to a broad range of systems beyond BECs, such as coherent electromagnetic fields and electron matter waves. Some basic concepts in spinor BECs and ultracold Fermi gases are mentioned. Two specific, presently timely, topics on two-dimensional vortex systems are discussed in some detail: (i) topological quantum computation with non-abelian vortex anyons, and (ii) two-dimensional quantum turbulence. The last two chapters list several physical systems where vortices play an important role and draw some phenomenological connections between certain concepts in high-energy particle physics, cosmology, and quantised vortices in superfluids. How to read this book This handbook is unlike traditional textbooks and is composed as a dictionarystyle electronic resource. It aims to serve multiple purposes, ranging from a quick reference for students to often-encountered concepts in physics of vortices in superfluids, to providing an interdisciplinary source of inspiration for career researchers interested in topological phenomena in ordered media. The intended use of this book involves reading it using an electronic device connected to the internet. Since the material is searchable electronically, inclusion of an index was deemed unnecessary. Instead, a more detailed table of contents that provides a quick overview of the whole material is provided with direct hyperlinks to the text. A return to the table of contents from anywhere within the document is meant to be easy using the abundantly provided hyperlinked Further Reading Suggestions (FRS), which serve as portals to the past and future citation light cones (CLCs). The past CLC of a citation entry comprises the list of references cited from that work; the future CLC of a citation entry comprises all works that appeared, or will appear, later and which cite the work in question. It is not the intention of this book to provide comprehensive lists of citations for any particular topic. Rather, one to a few arbitrarily selected citation events are provided for each dictionary entry that serve as an access point to the vast space–time of literature with overlapping CLCs. Readers are encouraged to post suggestions for the revised edition at https://sites.google.com/view/vortexbook. Selection of books • Fetter A L and Walecka J D 2003 Quantum Theory of Many-Particle Systems (New York: Dover). • Leggett A J 2006 Quantum Liquids (Oxford: Oxford University Press). xiii

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• Pethick C J and Smith H 2008 Bose–Einstein Condensation in Dilute Gases (Cambridge: Cambridge University Press). • Pitaevskii L and Stringari S 2016 Bose–Einstein Condensation and Superfluidity (Oxford: Oxford University Press). • Griffin A, Nikuni T and Zaremba E 2009 Bose-Condensed Gases at Finite Temperatures (Cambrdige: Cambridge University Press). • Ueda M 2010 Fundamentals and New Frontiers of Bose–Einstein Condensation (Singapore: World Scientific). • Gardiner C and Zoller P 2017 The Quantum World of Ultra-Cold Atoms and Light: Books I-III (Singapore: World Scientific). • Volovik G E 2009 The Universe in a Helium Droplet (Oxford: Oxford University Press). • Svistunov B, Babaev E and Prokof’ev N 2015 Superfluid States of Matter (Boca Raton, FL: CRC Press). • Proukakis N, Gardiner S, Davis M and Szymańska M (Eds) 2013 Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics (Singapore: World Scientific). • Parker N G and Barenghi C F 2016 A Primer on Quantum Fluids (Berlin: Springer). • Proukakis N P, Snoke D W and Littlewood P B (Eds) 2017 Universal Themes of Bose–Einstein Condensation (Cambridge: Cambridge University Press). • Donnelly R J 1991 Quantised Vortices in Helium II (Cambridge: Cambridge University Press). • Aftalion A 2006 Vortices in Bose–Einstein Condensates (Boston: Birkhäuser). • Wu J Z, Ma H-Y and Zhou M-D 2006 Vorticity and Vortex Dynamics (Berlin: Springer). • Nakahara M 2003 Geometry, Topology and Physics (Boca Raton, FL: CRC Press). • Nakahara M and Ohmi T 2008 Quantum Computing: From Linear Algebra to Physical Realizations (Boca Raton, FL: CRC Press). • Pachos J K 2012 Introduction to Topological Quantum Computation (Cambridge: Cambridge University Press). • Stanescu T D 2016 Introduction to Topological Quantum Matter and Quantum Computation (Boca Raton, FL: CRC Press). Selection of Review Articles and Monographs • Dalfovo F, Giorgini S, Pitaevskii L P and Stringari S 1999 Theory of Bose– Einstein condensation in trapped gases Rev. Mod. Phys. 71 463 • Leggett A J 2003 Bose–Einstein condensation in the alkali gases: Some fundamental concepts Rev. Mod. Phys. 73 307. • Giorgini S, Pitaevskii L P and Stringari S 2008 Theory of ultracold atomic Fermi gases Rev. Mod. Phys. 80 1215. • Fetter A L 2009 Rotating trapped Bose–Einstein condensates Rev. Mod. Phys. 81 647. xiv

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• Lahaye T, Menotti C, Santos L, Lewenstein M and Pfau T 2009 The physics of dipolar bosonic quantum gases Rep. Prog. Phys. 72 126401. • Blakie P B, Bradley A S, Davis M J, Ballagh R J and Gardiner C W 2008 Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques Adv. Phys. 57 363. • Mermin N D 1979 The topological theory of defects in ordered media Rev. Mod. Phys. 51 591. • Mizushima T, Tsutsumi Y, Kawakami T, Sato M, Ichioka M and Machida K 2016 Symmetry-protected topological superfluids and superconductors— from the basics to 3He J. Phys. Soc. Jpn. 85 022001. • Preskill J 2004 Lecture notes for physics 219: Quantum computation (accessed: 2018-04-19) • Nayak C, Simon S H, Stern A, Freedman M and Das Sarma S 2008 NonAbelian anyons and topological quantum computation Rev. Mod. Phys. 80 1083. • Field B and Simula T 2018 Introduction to topological quantum computation with non-Abelian anyons Quant. Sci. Technol. 3 045004. • Chu S 1998 Nobel Lecture: The manipulation of neutral particles Rev. Mod. Phys. 70 685. • Cohen-Tannoudji C N 1998 Nobel Lecture: Manipulating atoms with photons Rev. Mod. Phys. 70 707. • Phillips W D 1998 Nobel Lecture: Laser cooling and trapping of neutral atoms Rev. Mod. Phys. 70 721. • Cornell E A and Wieman C E 2002 Nobel Lecture: Bose–Einstein condensation in a dilute gas, the first 70 years and some recent experiments Rev. Mod. Phys. 74 875. • Ketterle W 2002 Nobel lecture: When atoms behave as waves: Bose–Einstein condensation and the atom laser Rev. Mod. Phys. 74 1131. • Abrikosov A A 2004 Nobel Lecture: Type-II superconductors and the vortex lattice Rev. Mod. Phys. 76 975. • Ginzburg V L 2004 Nobel Lecture: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the ‘physical minimum’ at the beginning of the XXI century Rev. Mod. Phys. 76 981. • Leggett A J 2004 Nobel Lecture: Superfluid 3He: the early days as seen by a theorist Rev. Mod. Phys. 76 999. • Kosterlitz J M 2017 Nobel Lecture: Topological defects and phase transitions Rev. Mod. Phys. 89 040501. • Haldane F D M 2017 Nobel Lecture: Topological quantum matter Rev. Mod. Phys. 89 040502. Other Selected Resources • The BEC Vortex Project: http://www.becvortex.com • NASA ADS: http://ui.adsabs.harvard.edu • arXiv.org: http://arxiv.org

xv

Acknowledgements Amy and Olli, thank you for the strawberries. I acknowledge funding from The Australian Research Council (ARC) via Grant Nos DP130102321, DP170104180, and FT180100020. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. Tapio Simula Aspen 2018

xvi

Author biography Tapio Simula Tapio Simula was awarded a DSc (Tech) degree in 2003 by the Helsinki University of Technology. His research interests include the physics of quantum vortices and superfluidity in Bose–Einstein condensates. He is currently an Australian Research Council (ARC) Future Fellow at Swinburne University of Technology, Melbourne, Australia.

xvii

Part I Vortices in Flatland

IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 1 Vortices

1.1 Space–time symmetries FRS: Noether's theorem1 The world around us is usually described as 3 + 1 dimensional, comprising three space dimensions, x , y, and z, and one time dimension, t. Lower dimensional physical systems in which the movement of the system’s constituents along one or more of the space–time dimensions are restricted, are embedded in this higher dimensional world. The symmetries of space and time, as formalised by Noether’s theorem, are associated with conservation laws of observable quantities. Space translation symmetry—the invariance of the system with respect to spatial translation—is associated with the conservation of linear momentum. Time translation symmetry—the invariance of the system with respect to passage of time—is associated with the conservation of energy. The most important symmetry in this treatise is the isotropy of the space. That the space appears the same in every direction leaves the system invariant with respect to rotations about a chosen axis and is associated with the conservation law of orbital angular momentum—an observable that underpins the existence of vortices. In a world with only one spatial dimension (1D), vortices cannot exist in the usual sense because the constituent fluid particles of the system cannot move around each other. In two spatial dimensions (2D) the constituent fluid particles of the system are able to swirl around a point without ever crossing that point itself, in which case this point represents a region of space embedded in the fluid that may be associated with a ‘particle’ that exists in the system due to the presence of a vortex in the fluid. On

1 E Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235 (1918).

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Figure 1.1. Appearance of the same vortex entity (the blue tubular curve) in one, two and three dimensional worlds. In 1D vortices cannot exist and appear as solitons, in 2D vortices appear as point particles, and in 3D vortices are string-like objects.

extending to three spatial dimensions (3D) such a vortex forms a string-like object about which the fluid particles swirl, as shown in figure 1.1. The electron may be viewed as an ‘elementary’ particle—a point embedded in the electromagnetic field comprised of a photon fluid. However, string theory posits that the electron would actually be a string object of a higher dimensional space–time but the human observers confined to live in the lower dimensional space–time are only able to perceive the point-like projection of such higher dimensional objects. Analogously, a Flatland observer will perceive the vortex line of 3D spaceland as an elementary point particle in 2D Flatland.

1.2 Quantum liquids FRS: quantum liquids2,3 A collection of molecules that are relatively free to move large distances through a fluid but typically remain in relatively close vicinity of the neighbouring fluid molecules serves as an intuitive picture of an incompressible classical fluid, such as water. To move a small patch of such a liquid imparts collective motion of a large number of other fluid particles. Contrasting with classical fluids, Leggett defines a quantum liquid as “a manyparticle system in whose behaviour not only the effects of quantum mechanics, but also those of quantum statistics are important”. Such quantum liquid behaviour becomes an important consideration for many-particle systems with high phasespace density

nλTD ≳ 1

2 3

A J Leggett, Quantum Liquids, Oxford University Press (2006). D Pines and P Nozières, The Theory of Quantum Liquids, CRC Press (2018).

1-2

(1.1)

Quantised Vortices

where n is the particle density, D is the dimension of the space and λT is the thermal de-Broglie wavelength

λT =

2π ℏ2 mkBT

(1.2)

that characterises the quantum mechanical ‘size’ of the particle with mass m at temperature T. The Boltzmann and Planck’s constants are denoted by kB, and ℏ, respectively. Quantum liquid behaviour is often associated with low-temperature phenomena such as superfluidity in liquid helium or superconductivity of metals and alloys. However, equations (1.1) and (1.2) show that equally high phase-space density conditions may be achieved at any temperature by isothermally compressing the fluid to higher density (if solidification can be avoided), or reducing the (effective) mass of the particles. Consequently, quantum liquid behaviour may also be expected in many ‘high-temperature’ systems including (i) the interiors of neutron stars, liquid metallic hydrogen in Jupiter’s interior, certain materials pressurised to gigapascal pressures using diamond anvils in a laboratory, due to high particle densities, and in (ii) trapped photon and exciton–polariton systems due to low effective particle mass.

1.3 Vorticity in classical fluids FRS: vortices in classical fluids4 A classical Newtonian fluid such as water can be settled into different types of rotational motion such as those illustrated in figure 1.2. If the fluid is placed in a bucket that is rotated at a constant angular frequency Ω, the fluid settles into equilibrium with the container and will rotate with angular frequency Ω as a rigid body whose vorticity

ω(r) = ∇ × v(r) = 2Ω

(1.3)

is uniformly distributed across the area of the fluid. The fluid’s velocity field v(r) therefore increases linearly with increasing radial distance from the centre of the bucket. If the bucket is not rotating but the water is being drained through a central hole in the bottom of the bucket the fluid develops a ‘bath-tub’ vortex state whose vorticity is concentrated near the centre of the vortex and vanishes at the walls of the bucket. The velocity field of such a vortex has two prominent regions: (i) in the core region the fluid velocity grows linearly as in a rigidly rotating body, (ii) outside the core region the fluid velocity is approximately inversely proportional to the distance from the centre of the bucket.

4

J Z Wu, H -Y Ma and M -D Zhou, Vorticity and Vortex Dynamics, Springer (2006).

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Figure 1.2. Rotation curves for classical fluids and superfluids.

1.4 Vorticity in quantum liquids FRS: vortices in superfluid helium5, vortices in Bose-Einstein condensates6 The velocity field of a simple superfluid characterised by a complex valued order parameter ψ (r) = ∣ψ ∣e iS (r) such as a cold atom Bose–Einstein condensate or a superfluid helium-4 is constrained to be curl-free and therefore cannot rotate as a true rigid body. Instead, such superfluids respond to forced rotation by spawning quantised vortices whose structure and behaviour has close similarities to the bathtub vortices of classical fluids (figure 1.2). The vorticity N

ω(r) = ∇ × ∇

ℏ ℏ v S (r) = ∑2πδ(r − rv), m m v=1

(1.4)

where m is the mass of the fluid particle, ℏ is the reduced Planck’s constant, and S (r) is the (classical action) velocity potential, is non-zero only at singular points rv that mark the locations of the Nv quantised vortices. Notwithstanding, a superfluid is able 5 6

R J Donnelly Quantised Vortices in Helium II, Cambridge University Press (1991). A L Fetter, Rotating trapped Bose–Einstein condensates, Reviews of Modern Physics 81, 647 (2009).

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Quantised Vortices

to mimic classical solid body rotation by distributing the individual quantum vortices uniformly across the fluid thus forming a vortex lattice that yields a spatially averaged vorticity field equivalent to classical solid body rotation (figure 1.2). This behaviour where the quantum system in the limit of large quantum numbers behaves in an average sense as a classical system may be viewed as an example of Bohr’s correspondence principle.

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IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 2 Quasiparticle picture

2.1 Emergence of quasiparticles FRS: Landau quasiparticles1, Bogoliubov quasiparticles2,3 Quasiparticles are emergent. They arise due to interactions between the constituent particles of the underlying physical system. As such, quasiparticles are a collective property of the system rather than an intrinsic property of the individual particles. It is debatable whether the so-called elementary or fundamental particles such as electrons are simply just another quasiparticle emerging out of an underlying layer of more elementary (quasi)particles. For example, a phonon is a quasiparticle of a crystal lattice and arises due to the collective behaviour of an assembly of the atoms. Spin waves in magnetic systems arise as a collective degree of freedom of the individual spins. A Cooper pair of electrons is a quasiparticle in a superconductor where the motion of individual electrons become pair-wise correlated due to the interaction with the environment they are embedded in. Many phases of matter can be described in terms of the picture of emergent quasiparticles whereby an interacting many-particle system is modelled in terms of a weakly or noninteracting system of quasiparticles. In superfluid systems the Bogoliubov quasiparticles are composite objects comprising amplitudes uq(r, t ) and vq(r, t ), where q is a label that uniquely identifies each quasiparticle. The quantum field ψˆ (r, t ) of the interacting parent system of particles may be expressed as a sum of the quasiparticles

1

G Baym and C J Pethick, Landau Fermi-Liquid Theory, Wiley-VCH (1991). N N Bogoliubov, On the Theory of Superfluidity, Journal of Physics (USSR) 11, 23 (1947). 3 A L Fetter, Nonuniform States of an Imperfect Bose Gas, Annals of Physics 70, 67 (1972). 2

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ψˆ (r , t ) =

∑uq(r , t )αˆ q ± vq*(r , t )αˆ q†,

(2.1)

q

where αˆq and αˆ q† are the quasiparticle creation and annihilation operators, respectively. The complex functions uq(r, t ) and vq(r, t ) that become hybridised in a quasiparticle are often referred to as the wave functions of the particle and hole (antiparticle) components of the quasiparticle, respectively. While the ‘elementary’ particles of a system may strongly interact, the quasiparticles are to first approximation non-interacting, which simplifies their treatment.

2.2 Boson commutation relations FRS: Boson commutation relations4 Bosonic quasiparticles are quantised according to the commutation relations

⎡⎣αq , α †⎤⎦ = δq,p p

and

⎡⎣α †, α †⎤⎦ = [αq , αp ] = 0. q p

(2.2)

For quasiparticles with bosonic quantum statistics, there is no limit on how many quasiparticles may simultaneously occupy the same state. Upon exchange of two bosons, the wave function

Ψ(r1, r2) = e iθ Ψ(r2 , r1)

(2.3)

describing such a system acquires a phase change of θ = 2π .

2.3 Fermion anticommutation relations FRS: Fermion anticommutation relations4 Fermionic quasiparticles are quantised according to the anticommutation relations

{αq, α p†} = δq,p

and

{αq†, α p†} = {αq, αp} = 0.

(2.4)

For quasiparticles with fermionic quantum statistics, the Pauli exclusion principle applies and it is forbidden for any two quasiparticles to occupy the same quantum state. Upon exchange of two fermions, the wave function

Ψ(r1, r2) = e iθ Ψ(r2 , r1)

(2.5)

describing such a system acquires a phase change of θ = π .

4

A L Fetter and J D Walecka, Quantum Theory of Many-Particle Systems, Dover Publications (2003).

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2.4 Majorana relations FRS: Majorana quasiparticles5 Majorana quasiparticles are quantised according to the relations

αq† = αq

and

{αq , αp} = 2δq,p.

(2.6)

As such, a Majorana quasiparticle and its antiparticle are equivalent, and creating two Majorana quasiparticles is equivalent to no particles at all (vacuum).

2.5 Anyon quasiparticles FRS: anyons6 In 2D systems (figure 2.1) quasiparticles are in general (abelian or non-abelian) anyons for which more general quantisation rules apply. However, the quantum statistics that interpolate between Bose–Einstein and Fermi–Dirac cases continue to be debated. Upon exchange of two abelian anyons, the wave function

Ψ(r1, r2) = e iθ Ψ(r2 , r1)

(2.7)

Figure 2.1. Geometric properties of anyons. (a) In 3D the closed (orange) path C around the blue particle is topologically equivalent to the (orange) path C′ and can be contracted to a single point p by ‘lifting’ the path over the particle utilising the third dimension. In 2D this is not possible. (b) Equivalence between moving the particle around a point p or moving a point around the particle. (c) Exchanging two identical particles, moving each of them half a loop, is equivalent to moving a single particle along an equivalent full closed path once. For non-abelian anyons also the direction of the path becomes important.

5

A J Leggett, Majorana fermions in condensed-matter physics, International Journal of Modern Physics B 30, 1630012 (2016). 6 F Wilczek, Quantum Mechanics of Fractional-Spin Particles, Physical Review Letters 49, 957 (1982).

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Quantised Vortices

describing such a system acquires a phase change of θ, which in general may take any value θ ∈ [0, 2π ].

2.6 Non-abelian anyon quasiparticles FRS: non-abelian anyons7 Upon exchange of two non-abelian anyons, the multicomponent wave function a

Ψμ(r1, r2) = e iθaTμν Ψν(r2 , r1)

(2.8)

describing such a system acquires a path-dependent phase change of θa , which in general is characterised by a non-commutative, with respect to different paths, matrix T. In such cases the outcome of exchanging two non-abelian anyons may depend on both the base point and the direction in which the particles are (braided) exchanged.

2.7 Bogoliubov–de Gennes equations FRS: Bogoliubov8-de Gennes9 equations, BdG symmetry classification10,11,12,13,14

The dynamics of the archetype quasiparticles may be described by the Bogoliubov– de Gennes (BdG) quasiparticle equations of motion

iℏ

⎛ u q (r , t ) ⎞ ∂ ⎛ u q (r , t ) ⎞ ⎟. ⎟ = M⎜ ⎜ ∂t ⎝ vq(r , t ) ⎠ ⎝ vq(r , t ) ⎠

(2.9)

For stationary states equation (2.9) yields an eigenvalue problem

⎛ u q (r) ⎞ ⎛ u q (r) ⎞ ⎟, ⎟ = Eq⎜ M⎜ ⎝ vq(r) ⎠ ⎝ vq(r) ⎠

7

(2.10)

F Wilczek, New Kinds of Quantum Statistics, Progress in Mathematical Physics 55, 61 (2009). N N Bogoliubov, On the Theory of Superfluidity, Journal of Physics (USSR) 11, 23 (1947). 9 P G de Gennes, Superconductivity of Metals and Alloys, Benjamin (1966). 10 A Altland and M R Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Physical Review B 55, 1142 (1997). 11 C -K Chiu, J C Y Teo, A P Schnyder, and S Ryu, Classification of topological quantum matter with symmetries, Reviews of Modern Physics 88, 035005 (2016). 12 T D Stanescu, Topological Quantum Matter and Quantum Computation, CRC Press (2016). 13 T Mizushima, Y Tsutsumi, T Kawakami, M Sato, M Ichioka, and K Machida, Symmetry-Protected Topological Superfluids and Superconductors—From the Basics to 3He, Journal of the Physical Society of Japan 85, 022001 (2016). 14 Z Gong, Y Ashida, K Kawabata, K Takasan, S Higashikawa, and M Ueda, Topological Phases of NonHermitian Systems, Physical Review X 8, 031079 (2018). 8

2-4

Quantised Vortices

where Eq is the excitation energy of the quasiparticle. The detailed form of the BdG matrix M depends on the pairing interactions of the particles. Possible forms include

⎛ E(r) Δ(r) ⎞ MFermi = ⎜ ⎟ ⎝Δ*(r) − E *(r)⎠

and

⎛ E(r) − Δ(r) ⎞ MBose = ⎜ ⎟, ⎝Δ*(r) − E *(r)⎠

(2.11)

where E(r) is a single-particle operator and Δ(r) is the operator characterising the pair correlations. The fermionic BdG matrix is Hermitian whereas the boson one is not. The quasiparticles of MFermi satisfy fermionic commutation relations. The quasiparticles of MBose satisfy bosonic commutation relations. Simple BECs may be modelled using a non-Hermitian BdG system of the MBose type. Conventional BCS superconductors, spin-polarised Fermi gases, and certain topological insulators such as the A-phase of superfluid helium-3 and chiral p-wave superconductors that feature Majorana quasiparticles may be modelled using a Hermitian BdG system of the MFermi type. Generically, Hermitian BdG systems may be classified into 10 symmetry groups in terms of the topological structure of their excitation spectrum; see table 2.1. For non-Hermitian systems these 10 classes may be organised into six groups according to table 2.2.

2.8 Time-reversal symmetry FRS: BdG symmetry classification10-14 The time-reversal operator T , which reverses the arrow of time t → − t, is an antiunitary operator that satisfies

T 2 = ±1.

(2.12)

Table 2.1. Periodic table of Hermitian Hamiltonians, for different relative dimensions δ, according to the Altland–Zirnbauer (AZ) classification.

Class

TRS

PHS

A AIII AI BDI D DIII AII CII C CI

0 0 1 1 0 −1 −1 −1 0 1

0 0 0 1 1 1 0 −1 −1 −1

SLS

δ=0

δ=1

δ=2

δ=3

δ=4

0 1 0 1 0 1 0 1 0 1

 0  2 2 0 2 0 0 0

0  0  2 2 0 2 0 0

 0 0 0  2 2 0 2 0

0  0 0 0  2 2 0 2

 0 2 0 0 0  2 2 0

2-5

Quantised Vortices

Table 2.2. Periodic table of non-Hermitian Hamiltonians, for different relative dimensions δ, expressed in terms of the Altland–Zirnbauer (AZ) classes.

AZ Class

δ=0

δ=1

δ=2

δ=3

δ=4

A, DIII, CI AIII AI,D BDI AII,C CII

0 0 2 2 ⊕ 2 0 0

 ⊕  ⊕ 2 2  ⊕ 2

0 0 0 0 0 0

 ⊕ 0 0 2 2 ⊕ 2

0 0 0 0 2 2 ⊕ 2

It can be expressed as a product T = UK of a unitary operator U and a complex conjugation operator K. For an integer spin particle T = K such that T 2 = K2 = 1. For a spin-½ particle the appropriate unitary operator is U = −iσy , where σy is the usual SU(2) Pauli matrix. The time-reversal operator for the fermionic system is therefore

T = −iσyK

(2.13)

and its square is T 2 = −1. The BdG matrix M is said to be time-reversal symmetric if

TMT † = M

(2.14)

is satisfied. Time-reversal symmetry with T 2 = ±1 is denoted by TRS = ±1. An absence of time-reversal symmetry is denoted by TRS = 0. For integer spin particles, the Hermitian BdG matrices have TRS = 1 if both E and Δ are real and TRS = 0 otherwise. The former is also the case for a simple scalar BEC (non-Hermitian) in its ground state and the latter is true if a quantised vortex Δ(r) ∝ e iθ is nucleated in such a system. These two cases belong to two different BdG symmetry classes and possess topologically distinct excitation spectra. Specifically, the presence of a topologically protected quantised vortex results in the emergence of core-localised kelvon modes in the BdG excitation spectrum. For spin-½ particles, TMFermi T † = −MFermi such that TRS = 0, whereas MBose has TRS = −1 if E = 0 and TRS = 0 otherwise.

2.9 Particle–hole symmetry FRS: BdG symmetry classification10-14 The particle–hole operator C , also known as charge-conjugation, which swaps particles and antiparticles, is an anti-unitary operator that satisfies

C 2 = ±1.

2-6

(2.15)

Quantised Vortices

It can be expressed as a product C = UK of a unitary operator U and a complex conjugation operator K. Two choices, respectively corresponding to C 2 = 1 and C 2 = −1, are

C = σxK

(2.16)

C = iσyK ,

(2.17)

and

where σα are the usual SU(2) Pauli matrices:

⎛ ⎞ σx = ⎜ 0 1 ⎟ , ⎝1 0 ⎠

⎛ ⎞ σy = ⎜ 0 − i ⎟ , ⎝i 0 ⎠

⎛ ⎞ σz = ⎜1 0 ⎟. ⎝ 0 − 1⎠

(2.18)

The BdG matrix M is said to be particle–hole symmetric if

CMC † = −C

(2.19)

is satisfied. Particle–hole symmetry with C 2 = ±1 is denoted by PHS = ±1. An absence of particle–hole symmetry is denoted by PHS = 0. For Δ = 0 the BdG matrix MFermi has particle–hole symmetry with PHS = 1 and for Δ ≠ 0 it has PHS = 0. The boson case MBose is particle–hole symmetric with PHS = 1. For Δ = 0 the BdG matrix MBose has particle–hole symmetry PHS = −1 and for Δ ≠ 0 it has PHS = 0. The BdG matrix MFermi is particle–hole symmetric with PHS = −1. For the BdG equations the particle–hole symmetry implies that the transformation

u q(r , t ) → vq*(r , t ),

vq(r , t ) → u q*(r , t ),

and

Eq → −Eq*

(2.20)

leaves the BdG equation invariant. The physic of these transformations is that for every quasiparticle eigenstate with energy Eq the quasiparticle excitation spectrum contains another state with energy −Eq*. For Fermi systems both modes are physically relevant whereas for Bose systems changing u’s and v’s changes the sign of the norm of the quasiparticle mode and therefore only one of the mode pair is considered physical.

2.10 Chiral symmetry FRS: BdG symmetry classification10-14 The sublattice operator S is an anti-unitary operator that satisfies

S2 = 1

(2.21)

and is expressed as a product S = TC of the time-reversal and particle–hole operators. The BdG matrix M is said to have chiral (or sublattice) symmetry if

2-7

Quantised Vortices

SMS † = −S.

(2.22)

Considering the sublattice operator

S = σx,

(2.23)

for real E and Δ = 0 the BdG matrix MBose has sublattice symmetry with SLS = 1 and otherwise SLS = 0. For real E and Δ the BdG matrix MBose has sublattice symmetry with SLS = 1 and otherwise SLS = 0.

2.11 Phonon spectrum FRS: Bogoliubov phonon15 For a simple BEC, the BdG equations have a plane wave solution with the spectrum

Eq =

⎛ p 2 ⎞2 p2 ⎜ q ⎟ + 2gn q , ⎜ 2m ⎟ 2m ⎝ ⎠

(2.24)

where

pq = ℏkq

(2.25)

is the linear momentum of a quasiparticle with a wave vector kq = 2π /λq and a corresponding wavelength λq . This spectrum smoothly interpolates between two extremes. For large momenta pq ≫ mgn or for non-interacting, g = 0, systems the spectrum corresponds to a quadratic free particle spectrum:

Eq =

pq2 2m

.

(2.26)

However, for low momenta, pq → 0, and with g ≠ 0 the spectrum is linear:

Eq = pq cs

(2.27)

and corresponds to waves propagating at the speed of sound:

cs =

gn . m

(2.28)

Such sound waves have the remarkable property that they also correspond to density modulations of the superfluid order parameter or the condensate wave function.

15

L Pitaevskii and S Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016).

2-8

Quantised Vortices

This is mathematically expressed by the statement that the poles of the singleparticle Green’s function and density–density response function are equal.

2.12 Landau critical velocity FRS: Landau critical velocity16,17 In a superfluid, an impurity may move through the superfluid without creating excitations in the fluid, which is one of the hallmarks of superfluid response. The maximum speed at which this remains true defines the Landau critical velocity. Above the critical velocity, vc , production of excitations is permitted by the conservation laws of energy and momentum. For the non-interacting ideal Bose gas with g = 0 in equation (2.24) the Landau critical velocity vanishes:

⎛ p 2 2m ⎞ ⎛E ⎞ q q ⎟ = 0, ⎟ ⎜ vc ≡ min ⎜ ⎟ = min ⎜⎜ ⎟ ⎝ pq ⎠ ⎝ pq ⎠

(2.29)

and the system is not superfluid, despite being a Bose–Einstein condensate (BEC). For a weakly interacting BEC with dispersion relation (2.24) the critical velocity equals the speed of sound:

⎛ p cs ⎞ ⎛E ⎞ q q ⎟⎟ = cs. vc ≡ min ⎜⎜ ⎟⎟ = min ⎜⎜ p ⎝ pq ⎠ ⎝ q⎠

(2.30)

The emergence of the linear part in the Bogoliubov spectrum (2.24) for low momenta is remarkable because it enables superfluid response. Such sound waves traveling at the speed of sound in a superfluid are an analog of massless photons in electrodynamics.

2.13 Roton–maxon spectrum FRS: roton-maxon spectrum18,19,20,21 The roton–maxon spectrum refers to a dispersion relation of the form 16

L D Landau, The theory of superfluidity of helium II, Journal of Physics USSR 5, 71 (1941). G Baym and C J Pethick, Landau critical velocity in weakly interacting Bose gases, Physical Review A 86, 023602 (2012). 18 L D Landau, On the theory of superfluidity of helium II, Journal of Physics USSR 11, 91 (1947). 19 R P Feynman, Atomic Theory of the Two-Fluid Model of Liquid Helium, Physical Review 94, 262 (1954). 20 L Santos, G V Shlyapnikov, and M Lewenstein, Roton–Maxon Spectrum and Stability of Trapped Dipolar Bose–Einstein Condensates, Physical Review Letters 90, 250403 (2003). 21 R J Donnelly, The Ghost of a Vanished Vortex Ring, Quantum Statistical Mechanics in the Natural Sciences 4, Springer (1974). 17

2-9

Quantised Vortices

E (p ) = Δ +

(p − p0 )2 , 2m*

(2.31)

where m* is the effective mass of the quasiparticle in consideration. The energy gap Δ = vcp0 equals the product of the Landau critical velocity vc and the momentum p0 of the roton minimum. The local maximum that occurs for p < p0 is referred to as the maxon. The roton–maxon spectrum was originally introduced to explain the low-energy excitations of a strongly interacting superfluid helium-4. Onsager is said to have viewed the roton quasiparticle as a tiny vortex ring whose radius has shrunk to be equal the radius of the vortex core—the ghost of a vanished vortex ring. In contrast to the superfluid helium, the usual Bogoliubov quasiparticle spectrum of simple weakly interacting BECs does not feature a roton minimum. However, sufficiently strong dipole–dipole interactions may induce a strong roton minimum also in weakly interacting BECs. The connection between such rotons in a weakly interacting Bose gas and a vortexonium (Jones–Roberts soliton) is an interesting topic of contemporary research.

2.14 Edge modes FRS: yrast state22, surface modes23,24 In finite-size systems the quasiparticles that are localized on the surface of the system are referred to as surface modes, surfons, or edge modes. For each mode branch with orbital angular momentum multipolarity ℓ , the mode with lowest principal quantum number n ℓ = 0 is a surface mode. The mode with the smallest excitation energy Eq per angular momentum ℓ is the yrast state. These surfons are particularly important in rotating BECs since they are the mediators of topological quantum phase transitions; see figure 2.2. For a non rotating system Ω = 0 in figure 2.2(a) the excitation spectrum is symmetric. When the system is subjected to a rotating drive (figure 2.2(b)) the spectrum becomes asymmetric as the excitations shift by ΔEq = −Ωℓ ℏ, in proportion to their angular momentum. When the Landau critical angular velocity Ωc = min(Eq /Ωℏℓq ) is reached (figure 2.2(c)) the yrast surfon becomes resonant with the Goldstone boson and triggers a change in the topology of the condensate via nucleation of a quantised vortex in the system. After the topological transition, the spectrum is left asymmetric (figure 2.2(d)) and new low-lying vortex core-localised excitation modes (kelvons) have emerged in the spectrum. For increasing rotation

22 B Mottelson, Yrast Spectra of Weakly Interacting Bose–Einstein Condensates, Physical Review Letters 83, 2695 (1999). 23 F Dalfovo and S Stringari, Shape deformations and angular-momentum transfer in trapped Bose–Einstein condensates, Physical Review A 63, 011601(R) (2000). 24 T P Simula, S M M Virtanen, and M M Salomaa, Surface modes and vortex formation in dilute Bose– Einstein condensates at finite temperatures, Physical Review A 66, 035601 (2002).

2-10

Quantised Vortices

Figure 2.2. Illustration of vortex nucleation in a rotated Bose–Einstein condensate (BEC) as a topological quantum phase transition. The red zero-energy mode is the Goldstone boson. The blue mode is the yrast state. The orange modes are vortex core-localised kelvon modes.

frequency, each new vortex nucleated in the system involves a closing of a gap and the emergence of an additional core mode associated with each new vortex.

2.15 Dipole, breathing, quadrupole and scissors modes FRS: collective modes25,26, Kohn mode27, scissors mode28,29, breathing mode30, quantum anomaly31,32,33 For a spherically symmetric 3D harmonically trapped BEC in the Thomas–Fermi limit the Bogoliubov excitation spectrum is

ω3D(n r , ∣ℓ∣) = ωosc 2n r2 + 2n r∣ℓ∣ + 3n r + ∣ℓ∣ ,

25

(2.32)

S Stringari, Collective Excitations of a Trapped Bose-Condensed Gas, Physical Review Letters 77, 2360 (1996). S Stringari, Dynamics of Bose–Einstein condensed gases in highly deformed traps, Physical Review Letters 58, 2385 (1998). 27 W Kohn, Cyclotron Resonance and de Haas–van Alphen Oscillations of an Interacting Electron Gas, Physical Review 123, 1242 (1961). 28 D Guéry-Odelin and S Stringari, Scissors Mode and Superfluidity of a Trapped Bose–Einstein Condensed Gas, Physical Review Letters 83, 4452 (1999). 29 T P Simula, M J Davis, and P B Blakie, Superfluidity of an interacting trapped quasi-two-dimensional Bose gas, Physical Review A 77, 023618 (2008). 30 L P Pitaevskii and A Rosch, Breathing modes and hidden symmetry of trapped atoms in two dimensions, Physical Review A 55, R853(R) (1997). 31 M Olshanii, H Perrin, and V Lorent, Example of a Quantum Anomaly in the Physics of Ultracold Gases, Physical Review Letters 105, 095302 (2010). 32 T Peppler, P Dyke, M Zamorano, I Herrera, S Hoinka, and C J Vale, Quantum Anomaly and 2D–3D Crossover in Strongly Interacting Fermi Gases, Physical Review Letters 121, 120402 (2018). 33 M Holten, L Bayha, A C Klein, P A Murthy, P M Preiss, and S Jochim, Anomalous Breaking of Scale Invariance in a Two-Dimensional Fermi Gas, Physical Review Letters 121, 120401 (2018). 26

2-11

Quantised Vortices

where ωosc is the harmonic oscillator frequency and nr and ℓ are the principal and orbital angular momentum quantum numbers, respectively. In a purely 2D trap the frequencies are

ω 2D(n r , ∣ℓ∣) = ωosc 2n r2 + 2n r∣ℓ∣ + 2n r + ∣ℓ∣ .

(2.33)

The doubly degenerate dipole mode with nr = 0 and ℓ = ±1 is also known as the Kohn mode or the slosh mode and is the lowest excited state in the system with an excitation energy ℏωosc . Corresponding to the displacement of the centre of mass of the whole system, the energy of this mode is unaffected by particle interactions or the presence of thermal cloud. In a system with an axisymmetric vortex that is rotated with angular speed Ω the degeneracy of the two dipole modes is lifted such that the difference Eℓ=+1 − Eℓ=−1 = 2ℏΩ between their energies is determined by the rotation frequency alone. The breathing mode with nr = 1 and ℓ = 0 is the lowest excited state with zero angular momentum. It is revealed as a radially symmetric expansion and compression of the radius of the condensate. In a 2D harmonically trapped system the breathing mode is universal and occurs at 2ℏωosc independent of the strength of the particle interactions and thermal atoms reflecting a hidden SO(2,1) symmetry of the Hamiltonian. If the particle interactions are not of the contact type but have a long-range tail, the breathing mode is predicted to shift away from the universal value of 2ℏωosc , a phenomenon referred to as a quantum anomaly. With sufficiently strong resonant driving, the breathing mode with Eq = 2ℏωosc and ℓ = 0 may decay into a pair of dipole modes Eq = ℏωosc and ℓ = ±1 in an energy and angular momentum conserving process ultimately resulting in a spontaneous symmetry breaking and vortex nucleation. The quadrupole surface mode with nr = 0 and ℓ = ±2 plays an important role in vortex nucleation and can be used for resonant excitation of Kelvin waves of a vortex line. In an anisotropic trap, the quadrupole operator may be used to excite the so-called scissors mode, which is sensitive to the moment of inertia of the system and may be used for detecting the BKT-type superfluid–normal transitions.

2.16 Kelvin mode vortex waves FRS: Kelvin waves34, Kelvin mode35, kelvons36, experimental evidence37,38

34

W Thomson, Vibrations of a columnar vortex, Philosophical Magazine 10, 155 (1880). A L Fetter, Kelvin mode of a vortex in a nonuniform Bose–Einstein condensate, Physical Review 69, 043617 (2004). 36 T P Simula, T Mizushima and K Machida, Kelvin Waves of Quantized Vortex Lines in Trapped Bose– Einstein Condensates, Physical Review Letters 101, 020402 (2008). 37 V Bretin, P Rosenbusch, F Chevy, G V Shlyapnikov and J Dalibard, Quadrupole Oscillation of a SingleVortex Bose–Einstein Condensate: Evidence for Kelvin Modes, Physical Review Letters 90, 100403 (2003). 38 E Fonda, D P Meichle, N T Ouellette, S Hormoz and D P Lathrop, Direct observation of Kelvin waves excited by quantized vortex reconnection, PNAS 111, 4707 (2014). 35

2-12

Quantised Vortices

A Kelvin wave in a 3D system is a helical displacement wave of the centre line of the vortex that propagates along the length of the vortex. William Thomson (Lord Kelvin) obtained the dispersion relation for such waves:

⎡ K 0(krc ) ⎤ ⎞ 1 + krc⎢ ⎥ ⎟, ⎣ K1(krc ) ⎦ ⎟⎠

⎛ Γ ⎜ ωK = 1− 2πrc2 ⎜⎝

(2.34)

where k is the wave vector, Γ is the circulation of the vortex with core size rc, and Kj is a modified Bessel function of order j. This has a long wavelength, k → 0, asymptotic form:

ω KL =

⎤ Γk 2 ⎡ ⎛ 2 ⎞ ⎢ln ⎜ ⎟ − γ ⎥ , 4π ⎣ ⎝ krc ⎠ ⎦

(2.35)

where γ is the Euler–Mascheroni constant. Joseph Thomson considered slowly varying Kelvin waves on a vortex ring finding a dispersion relation

ω KR =

ηVR , R

(2.36)

where VR is the speed of propagation of the vortex ring and 1 η = p2 (p2 − 1) ≈ p2 − 2 is expressed in terms of the integer p = 1, 2, 3…. In superfluids, a quantised vortex is a host to a kelvon quasiparticle, which is a low-energy elementary excitation and the fingerprint of the vortex. The vortex core forms a quadratic confining potential for the kelvon, which is a bound state localised in the vortex core. The excitation of such vortex core-localised bound states manifests in a manner similar to Kelvin waves on classical vortices. Direct numerical calculations using the BdG equations find the dispersion relation for the kelvons:

ωqBdG = ω0 +

ℏk q2 ⎡ ⎛ 1 ⎞⎤ ⎢ln ⎜ ⎟⎥ , 2M ⎢⎣ ⎝ kqrc ⎠⎥⎦

(2.37)

where the zero-point frequency ω0 is related to the uncertainty in the vortex position. In 2D systems the kelvon corresponds to cyclotron motion of a charged particle. For harmonically trapped 2D systems ω0 has a Thomas–Fermi estimate:

ω 0TF = −

⎛R ⎞ 3ℏ ln ⎜ ⊥ ⎟ , 2 2mR ⊥ ⎝ rc ⎠

(2.38)

where m is the mass of the fluid particle and R⊥ is the Thomas–Fermi radius of the condensate. The kelvon determines the inertial mass of the vortex and is also important for nucleation of vortices and precessional motion of the vortex in trapped condensates.

2-13

Quantised Vortices

2.17 Tkachenko mode vortex waves FRS: Tkachenko waves39,40,41,42,43, direct detection44 In a vortex lattice with Nv vortices there are Nv branches of low-energy excitation modes. In the large vortex lattice limit the Tkachenko mode has the lowest excitation energy and corresponds to torsional motion of the whole vortex lattice where individual vortices orbit along elliptical trajectories around their own unperturbed positions. In rapidly rotating systems the Tkachenko mode is predicted to become unstable leading to the melting of the vortex lattice. Considering a compressible Bose gas, Baym obtained a Tkachenko dispersion relation

ω T2(k ) =

2C2 s 2k 2 , nm 4Ω2 + [s 2 + 4(C1 + C2 ) nm ]k 2

(2.39)

where n is the condensate density, s is the speed of sound, C1 is the compressional modulus and C2 is the shear modulus of the vortex lattice. For incompressional fluids C1 = −C2 = −Ωn/8. The dispersion relation (2.39) has two limits determined by the rigidity of the vortex lattice. In the stiff limit the waves are sound waves linear in momentum:

ω Tstiff (k ) =

Ω k, 4m

(2.40)

whereas in the soft limit the dispersion relation

ω Tsoft(k ) =

C2 s 2 k 2nm Ω

(2.41)

becomes quadratic.

39

V K Tkachenko, Stability of Vortex Lattices, Soviet Physics JETP 23, 1049 (1966). G Baym, Tkachenko Modes of Vortex Lattices in Rapidly Rotating Bose–Einstein Condensates, Physical Review Letters 91, 110402 (2003). 41 T P Simula, A A Penckwitt, R J Ballagh, Giant Vortex Lattice Deformations in Rapidly Rotating Bose– Einstein Condensates, Physical Review Letters 92, 060401 (2004). 42 T Mizushima, Y Kawaguchi, K Machida, T Ohmi, T Isoshima, M M Salomaa, Collective Oscillations of Vortex Lattices in Rotating Bose–Einstein Condensates, Physical Review Letters 92, 060407 (2004). 43 T Simula, Collective dynamics of vortices in trapped Bose-Einstein condensates, Physical Review A 87, 023630 (2013) 44 I Coddington, P Engels, V Schweikhard, E A Cornell, Observation of Tkachenko Oscillations in Rapidly Rotating Bose–Einstein Condensates, Physical Review Letters 91, 100402 (2003). 40

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Quantised Vortices

2.18 Caroli–de Gennes–Matricon modes FRS: CdGM modes45,46,47 In Fermi systems, the vortex core-localized quasiparticles that are the counterpart to Kelvin modes of Bose systems are known as Caroli–de Gennes–Matricon (CdGM) modes and may be viewed as Andreev bound states trapped by the normal state vortex core. Whereas for a single vortex in a 2D BEC there exists only one kelvon mode, the Fermi vortex core is able to host multiple bound states. The CdGM modes are low-energy excitations that exist within the superconducting gap. In s-wave BCS superconductors these CdGM modes have a level spacing: 1 Δ2 (2.42) , ω0 ≈ 2 EF where Δ is the gap function and EF is the Fermi energy, and the CdGM spectrum is

⎛ 1⎞ E ℓBCS = −⎜ℓ − ⎟ω0, ⎝ 2⎠

(2.43)

where ℓ is the integer angular momentum quantum number. In spin-polarized chiral p-wave superfluids in the BCS regime the CdGM spectrum is

⎛ ⎛ w + 1⎞ w + 1⎞ ⎟ω0 + ⎜n − ⎟ω1, Eℓ,n,w = −⎜ℓ − ⎝ ⎠ ⎝ 2 2 ⎠

(2.44)

where n is an integer and ω1 ≈ π2 Δ is proportional to the gap energy. The spectrum (2.44) interpolates between the near-Fermi energy and bulk excitations and has the special property that for a vortex with any odd winding number w = ±1, ±3, … the system has an exact zero-energy state with orbital angular momentum ℓ = (w + 1)/2. These are Majorana zero modes, which are topologically protected via coupling to a zero-energy edge mode.

2.19 Nambu–Goldstone zero mode FRS: Nambu-Goldstone boson48,49 Nambu-Goldstone zero mode50

45 C Caroli, P G De Gennes and J Matricon, Bound Fermion states on a vortex line in a type II superconductor, Physics Letters 9, 307 (1964). 46 N Hayashi, T Isoshima, M Ichioka, and K Machida, Low-Lying Quasiparticle Excitations around a Vortex Core in Quantum Limit, Physical Review A 80, 2921 (1998). 47 T Mizushima and K Machida, Vortex structures and zero-energy states in the BCS-to-BEC evolution of p-wave resonant Fermi gases, Physical Review A 81, 053605 (2010). 48 Y Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity Physical Review 117, 648 (1960). 49 J Goldstone, Field theories with ‘Superconductor’ solutions Il Nuovo Cimento 19, 154 (1961). 50 M Nitta and D A Takahashi, Quasi-Nambu–Goldstone modes in nonrelativistic systems Physical Review D 91, 025018 (2015).

2-15

Quantised Vortices

The Nambu–Goldstone boson zero mode is a quasiparticle which is its own antiparticle and has precisely zero energy, E0 = E − μ = 0, with respect to the chemical potential μ. The quasiparticle amplitudes satisfy the relation uq(r, t ) = vq*(r, t ) = ϕ(r, t ), where the condensate wave function ϕ(r, t ) is the solution to the Gross–Pitaevskii equation. Generically, the Goldstone boson is associated with the concept of spontaneous breaking of a continuous symmetry. For example, as the U(1) symmetry of the original Hamiltonian describing cold Bose gas becomes broken and the newly formed BEC acquires a spatial phase coherence, a quasiparticle mode emerges at zero energy in the Bogoliubov excitation spectrum and the phase of this quasiparticle rotates at the frequency of the chemical potential. As such, this mode acts to restore the broken U(1) symmetry by sampling all phase values in a timeaveraged sense. The new quasiparticle associated with this symmetry breaking is the massless Goldstone boson of the theory—the BEC itself.

2.20 Majorana zero mode FRS: Majorana fermion51, Majorana zero mode52,53,54,55 The Majorana fermion zero mode is a quasiparticle which is its own antiparticle in the sense that its creation and annihilation operators, η† = η, are equal. It occurs precisely at zero energy, E0 = E − EF = 0, with respect to the Fermi energy EF and satisfies uq(r, t ) = vq*(r, t ). Generically, a single fermion may be expressed as a sum of two Majoranas since the regular fermion operators α† and α that obey the anticommutation relations

{αi†, αj} = δij ,

{αi†, α j†} = {αi , αj} = 0

(2.45)

can always be expressed in terms of the Majorana operators ηj1 and ηj2 as a superposition:

αj =

1 (ηj1 + iηj 2 ), 2

α j† =

1 (ηj1 − iηj 2 ), 2

(2.46)

where the Majorana operators satisfy

{ηim , ηjn} = 2δijδmn

and

51

η jn† = ηjn.

(2.47)

E Majorana, Teoria simmetrica dell’elettrone e del positrone Il Nuovo Cimento 14, 171 (1937). A J Leggett, Majorana fermions in condensed-matter physics International Journal of Modern Physics B 30, 1630012 (2016). 53 S R Elliott and M Franz, Colloquium: Majorana fermions in nuclear, particle, and solid-state physics Reviews of Modern Physics 87, 137 (2015). 54 V Gurarie and L Radzihovsky, Zero modes of two-dimensional chiral p-wave superconductors Physical Review B 75, 212509 (2007). 55 T Mizushima and K Machida, Vortex structures and zero-energy states in the BCS-to-BEC evolution of pwave resonant Fermi gases Physical Review A 81, 053605 (2010). 52

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The Majorana zero modes are special because in that case the two Majoranas comprising the single fermionic excitation can be spatially separated becoming unpaired Majoranas:

ηj1 = α j† + αj

and

(

ηj 2 = i α j† − αj

)

(2.48)

that may be viewed as superpositions of a particle and a hole. For example, in a 2D topological superfluid such as a spin-polarised, chiral p-wave Fermi gas in the presence of a vortex, a zero-energy-vortex, core-localized CdGM mode emerges in the Bogoliubov excitation spectrum and is hybridised with a zero-energy edge mode localised at the surface of the superfluid. Such Majorana zero-mode quasiparticles are thought to be useful for topological quantum information processing.

2.21 Magnon spin waves FRS: magnons56,57,58,59,60 In spinor condensates the spin degree of freedom, the spatio-temporal orientation of the local spin vector, may vary resulting in spin waves whose quanta are called magnons. In a scalar BEC the broken U(1) symmetry results in a low-energy phonon branch with a zero-energy Goldstone boson. In a spinor BEC the SO(3) symmetry is also broken which results in additional low-energy spin-wave magnon branches. Generically, a spin-F BEC may have up to 2F + 1 low-lying phonon and magnon branches. Each of these mode branches are of the linear (phonon-like) or quadratic (single-particle-like) form

Eq = pq cs + Δm

or

Eq =

pq2 2m*

+ Δm ,

(2.49)

where pq is the momentum of the excitation, cs is the speed of sound, Δm is the excitation gap and m* is the effective mass of the quasiparticle. The value of the energy gap Δm depends on the details, in particular the coupling constants, of the ground state in question and may either be zero (gapless spectrum) or non-zero (gapped spectrum).

56

T -L Ho, Spinor Bose Condensates in Optical Traps, Physical Review Letters 81, 742 (1998). T Fukuhara, P Schauß, M Endres, S Hild, M Cheneau, I Bloch and C Gross, Microscopic observation of magnon bound states and their dynamics, Nature 502, 76 (2013). 58 G E Marti, A MacRae, R Olf, S Lourette, F Fang, and D M Stamper-Kurn, Coherent Magnon Optics in a Ferromagnetic Spinor Bose–Einstein Condensate, Physical Review Letters 113, 155302 (2014). 59 D Baillie and P B Blakie, Spin-dependent Bragg spectroscopy of a spinor Bose gas, Physical Review A 93, 033607 (2016). 60 Y Kawaguchi and M Ueda, Spinor Bose–Einstein condensates, Physics Reports 520, 253 (2012). 57

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IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 3 Cold atoms

3.1 Scalar Bose–Einstein condensates FRS: Bogoliubov theory1,2,3 Scalar BECs may be modelled using the Bogoliubov–de Gennes matrix MBose . Within the Hartree–Fock–Bogoliubov approximation in the rotating frame the diagonal and off-diagonal terms are, respectively,

E(r) = −

ℏ2 2 ∇ − μ + Vext(r , t ) + g∣ψ (r , t )∣2 + 2gρ(r , t ) − ΩLz , 2m

(3.1)

and

Δ(r) = ψ 2(r) + m ˜ (r).

(3.2)

In equation (3.1), m is the mass of a condensate particle, μ is the chemical potential, Vext(r, t ) is an external potential such as a harmonic or uniform trap, the coupling constant 4π ℏ2a (3.3) g= , m with a the s-wave scattering length, determines the strength of (g > 0 repulsive, g < 0 attractive) contact interactions, Ω is the angular frequency of the rotating

1

A L Fetter, Nonuniform states of an imperfect bose gas, Annals of Physics 70, 67 (1972). A Griffin, Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures, Physical Review B 53, 9341 (1996). 3 C W Gardiner, Particle-number-conserving Bogoliubov method which demonstrates the validity of the timedependent Gross–Pitaevskii equation for a highly condensed Bose gas, Physical Review A 56, 1414 (1997). 2

doi:10.1088/2053-2571/aafb9dch3

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ª Morgan & Claypool Publishers 2019

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frame, and Lz is the z-component of the orbital angular momentum operator. The BEC is an exact zero-energy Goldstone mode of the BdG equations, which satisfies the Gross–Pitaevskii (GP) equation

i ℏ∂tψ (r , t ) = E(r)ψ (r , t ) + gΔ(r , t )ψ *(r , t ).

(3.4)

This can be shown by substituting Eq = 0 and uq(r) = vq*(r) = ψ (r) into the BdG equations. The linearised perturbations to the condensate ground state may be expressed as

ψ (r , t ) = ψ0(r) + u q(r)e−iωqt + vq*(r)e iωqt.

(3.5)

Substitution of equation (3.5) into the GPE yields the Bogoliubov equations:

E(r)u q(r) − gψ 2(r)vq(r) = Equ q(r)

(3.6)

E *(r)vq(r) − gψ *2(r)u q(r) = −Eqvq(r)

(3.7)

for the linearised perturbations. In equilibrium, the self-consistent potentials ρ(r) and ‘gapful’ m˜ (r) are

ρ(r) =

∑ f (Eq ){∣uq(r)∣2

+ ∣vq(r)∣2 } + ∣vq(r)∣2

(3.8)

q

and

m ˜ (r) = −∑{2f (Eq ) + 1}vq*(r)up(r),

(3.9)

q

where

f (Eq ) = 1 (e Eq

k BT

− 1)

(3.10)

is the Bose–Einstein distribution function with kB the Boltzmann constant and T the temperature. The normalisation conditions for the Eq ≠ 0 quasiparticle amplitudes are

∫ {u q*(r)up(r) − vq*(r)vp(r)}d r = δqp

(3.11)

∫ {uq(r)vp(r) − up(r)vq(r)}d r = δqp.

(3.12)

and

The zero modes may be normalised according to

∫ ∣uq(r)∣2 d r = ∫ ∣vq(r)∣2 d r.

(3.13)

The excited quasiparticle states are not strictly orthogonal to the condensate mode and should be re-weighted to satisfy the orthogonality criterion. However,

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for large-particle-number systems this effect becomes negligible. The entropy in terms of the bosonic quasiparticles is

SB = kB ∑{(1 + fq ) ln(1 + fq ) − fq ln( fq )},

(3.14)

q

where fq = f (Eq ) is the equilibrium quasiparticle distribution function.

3.2 Bose zero-temperature energy functional FRS: ground state properties4, theory of BEC5,6 The energy functional for a zero-temperature Bose gas with contact interactions is

E = Ekin + E pot + E int =

⎛

⎞

2

∫ ⎜⎝ 2ℏm ∣∇ψ (r)∣2 + V ∣ψ (r)∣2 + 12 g∣ψ (r)∣4 ⎟⎠ d Dr,

(3.15)

where the condensate wave function is normalized to the condensate particle number

N=

∫ ∣ψ (r)∣2 d Dr.

(3.16)

Minimizing the variation of the energy with respect to ψ * yields the time-dependent GP equation:

⎞ δE ∂ψ (r) ⎛ ℏ2 2 = ⎜− ∇ + V (r) + g∣ψ (r)∣2 ⎟ ψ (r) = HGPψ (r). = iℏ ⎝ 2m ⎠ δψ *(r) ∂t

(3.17)

The GP equation may also be informally obtained from the Heisenberg equation of motion:

iℏ

∂ψˆ (r , t ) = [ψˆ (r , t ), Hˆ ], ∂t

(3.18)

with the replacements Hˆ = HGP and ψˆ = ψ (r, t ). The ground state of a harmonically trapped condensate satisfies a virial theorem:

2Ekin − 2E pot + 3E int = 0

(3.19)

and has the chemical potential

μ = (Ekin + E pot + 2E int ) N .

(3.20)

4 G Baym and C J Pethick, Ground-State Properties of Magnetically Trapped Bose-Condensed Rubidium Gas, Physical Review Letters 76, 6 (1996). 5 C J Pethick and H Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press (2008). 6 L Pitaevskii and S Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016).

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3.3 Thomas–Fermi relations FRS: basic relations4-6 In the limit (N → ∞) of a large number of condensate atoms, the quantum pressure (the quantum kinetic energy due to the Laplacian term) for a ground-state condensate described by the GP equation, equation (3.17), may be neglected enabling many analytical results to be obtained. The Thomas–Fermi (TF) condensate density for a harmonic trap is

nTF(r) = ∣ψ (r)∣2 =

⎛ μ − V (r) = n 0 ⎜⎜1 − g ⎝

∑ α

xα2 ⎞ ⎟, R α2 ⎟⎠

(3.21)

where the TF radii are

Rα =

2μ . mωα2

(3.22)

bμ , m

(3.23)

The speed of sound is

cs =

where the parameter b ≈ 1 accounts for the inhomogeneous condensate density. The chemical potential μ = ∂∂NE is 2

3D μTF

1

2

⎛ 15 ⎞ 5 ⎛ gN ⎞ 5 =⎜ ⎟ ℏω ⎟ ⎜ ⎝ 512 π ⎠ ⎝ ℏωa 03 ⎠

and

2D μTF

1 ⎛ 1 ⎞ 2 ⎛ g 2DN ⎞ 2 =⎜ ⎟ ⎜ ⎟ ℏω, ⎝ π ⎠ ⎝ ℏωa 02 ⎠

where g 2D = g / 2π lz is the effective 2D coupling constant and a 0 = harmonic oscillator length.

(3.24)

ℏ/mω is the

3.4 Healing length FRS: basic relations4-6 The healing length is a characteristic length scale over which the condensate wave function heals to its bulk value when locally forced to zero. Considering the point at which the kinetic energy term is in balance with the interaction energy term

ℏ2 2 ∣∇ ψ (r)∣ = gn(rc ) 2m rc motivates the definition for the healing length as

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(3.25)

Quantised Vortices

ξ=

ℏ2 = 2mgn

ℏ . 2mμ

(3.26)

In a scalar BEC the healing length is typically of the order of the vortex core radius. In non-interacting or rapidly rotating systems the relationship between the healing length and the vortex core size is lost since the interaction energy term becomes vanishingly small in such situations and the constraint for the vortex core size is instead determined by the quantum pressure and vortex density.

3.5 Thermodynamic relations FRS: trapped ideal gas7, theory of BECs5,6 Bose–Einstein condensation is expected when the phase-space density D nλ dB ⩾ 1,

(3.27)

where

λ dB =

h 2πmkBT

(3.28)

is the thermal de Broglie wavelength and D is the space dimension, is sufficiently high such that the wave functions of the particles overlap. The critical temperature and condensate fraction of a uniform ideal Bose gas are

kBTc3D =

ℏ2 ⎛ N ⎞2 3 2π ⎜ ⎟ ζ(3 2)2 3 m ⎝V ⎠

and

⎛ T ⎞3 N0 =1−⎜ ⎟. ⎝ Tc ⎠ N

(3.29)

In harmonically trapped 1D, 2D and 3D systems, respectively, with harmonic oscillator frequency ω0, the critical temperature and the condensate fraction are

kBTc1D = ℏω0N ;

kBTc2D = ℏω0N1 2;

kBTc3D = ℏω0N1 3

(3.30)

⎛ T ⎞2 N0 =1−⎜ ⎟ ; ⎝ Tc ⎠ N

⎛ T ⎞3 N0 =1−⎜ ⎟. ⎝ Tc ⎠ N

(3.31)

and

⎛ T ⎞1 N0 =1−⎜ ⎟; ⎝ Tc ⎠ N

The equation of state of a uniform weakly interacting BEC, expressed in terms of the pressure P, equals P = gn2 /2, which is a polytropic equation of state

7 V Bagnato, D E Pritchard and D Kleppner, Bose–Einstein condensation in an external potential 35 4354 (1987).

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1

P = cpn1+ p

(3.32)

with polytropic index p = 1 and polytropic constant cp = g/2. The chemical potential is μ = gn such that the Gibbs–Duhem relation

ndμ = −sdT + dP

(3.33)

is satisfied at zero temperature where n = N/V and s = S/V are the particle and entropy densities and m is the mass of a particle.

3.6 Quantum hydrodynamic equations FRS: Madelung equations8 The Madelung form of the complex valued wave function, ψ (r) = ∣ψ (r)∣e iS (r), may be used for expressing Schrödinger-like equations, such as the GP equation, in a quantum hydrodynamic form. After substitution and separating the real and imaginary parts, the GPE yields the continuity equation ∂n (3.34) , ∇ · (nvs ) = ∂t and an Euler-like equation:

m

⎛1 ⎞ ℏ2 ∇2 n ∂vs = −∇ ⎜ mvs2 − + gn + Vext⎟. ⎝2 ⎠ 2m ∂t n

(3.35)

Solving these two equations with the constraint that ωs = ∇ × vs = 0 everywhere where ψ (r) is non-zero is, in principle, equivalent to solving the GPE directly. However, the irrotationality condition for the hydrodynamical equations may be challenging to implement. The curl of the Euler equation yields the vorticity equation

∂ωs = ∇ × (vs × ωs ) = (ωs · ∇)vs − (vs · ∇)ωs , ∂t

(3.36)

where the superfluid velocity field vs = ℏ∇S (r)/m may have singularities. The vortexstretching terms on the right account for the bending deformations of the vortex lines due to the velocity gradients and they are absent in 2D flows. Therefore, for an incompressible 2D inviscid flow the vorticity is a constant of motion:

∂ωs = 0. ∂t

(3.37)

This may also be expressed in the form of Kelvin’s circulation theorem:

8 E Madelung, Eine anschauliche Deutung der Gleichung von Schrödinger, Die Naturwissenschaften 14 1004 (1926).

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Γ=

∫C(t ) vs · dℓ = const,

(3.38)

which states that the vortices in such systems are ‘frozen’ and move with the patches of fluid, bounded by the curve C(t ), they are embedded in. This statement is mathematically equivalent to Liouville’s theorem of statistical mechanics, which states that the volume of patches of phase space is preserved irrespective of how their shape is deformed. Here it is worth remembering the equality of the (x , y ) real space in which the point-like vortices of the fluid move and the (x , px ) phase space of the 1D particles whose phase-space dynamics the vortex particles represent.

3.7 Two-component Bose–Einstein condensates FRS: condensate mixtures9 For two-component BECs, comprising for example two atomic species, isotopes or hyperfine states, the dimension of the order parameter is doubled and may be described as a pseudospin-1/2 system. The condensate ground state may be described by a two-component GP equation:

i ℏ∂tϕ1(r , t ) = H1(r)ϕ1(r , t ) + {g11∣ϕ1(r , t )∣2 + g12∣ϕ2(r , t )∣2 }ϕ1(r , t ) − ℏΩRϕ2(r , t ) i ℏ∂tϕ2(r , t ) = H2(r)ϕ2(r , t ) + {g22∣ϕ2(r , t )∣2 + g21∣ϕ1(r , t )∣2 }ϕ2(r , t ) − ℏΩRϕ1(r , t ),

(3.39)

where the single-particle Hamiltonians are

Hj (r) = −

ℏ2 2 ∇ − μj + Vext(r , t ) − ΩLz. 2mj

(3.40)

In equations (3.39), gij is a matrix of inter- and intra-component density-dependent coupling constants and the scenario-dependent coupling constants ΩR(t ) represent phase-dependent coherent (Rabi) coupling between the two condensate components. Generalisation to N component condensate mixtures generically results in a set of N-coupled GP equations. Depending on the values of the coupling constants, the stable condensate ground states may be miscible or immiscible with the richness of possible ground-state phases increasing with the number of condensate components.

9

C J Pethick and H Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press (2008).

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Figure 3.1. Spinor phase diagrams for spin-1 BECs with antiferromagnetic (left) and ferromagnetic (middle) spin–spin coupling constants. The frame in the right is for spin-2 BEC in the absence (B = 0) of external magnetic fields. The labels refer to ferromagnetic (FM), antiferromagnetic (AF), polar (P), broken axisymmetry (BA), cyclic-tetrahedral (CT), uniaxial nematic (UN) and biaxial nematic (BN).

3.8 Spin-1 Bose–Einstein condensates FRS: spinor BECs10,11,12,13 A zero-temperature ground-state phase diagram of spin-1 BEC in the presence of an external magnetic field that produces linear and quadratic Zeeman shifts of strengths p and q, respectively, is shown in figure 3.1(left) for antiferromagnetic spin–spin interactions, c1 > 0, and in figure 3.1(middle) for ferromagnetic spin–spin interactions, c1 < 0. Depending on the values for c0 = 4π ℏ2(a 0 + 2a2 )/3m and c1 = 4π ℏ2(a2 − a 0 )/3m and the Zeeman coupling constants, a number of ground-state phases such as the ferromagnetic, antiferromagnetic, polar or broken axisymmetry, can be realised. The mean-field Hamiltonian density is c c H (r) = T (r) + Vext(r) + 0 n(r)2 + 1 ∣F (r)∣2 + (q 2 − p )Fz(r), (3.41) 2 2 where the single-particle kinetic energy density is

10

T -L Ho, Spinor Bose Condensates in Optical Traps, Physical Review Letters 81, 742 (1998). T Ohmi and K Machida, Bose–Einstein condensation in an external potential, Journal of the Physical Society of Japan 67, 1822 (1998). 12 D M Stamper-Kurn and M Ueda, Spinor Bose gases: Symmetries, magnetism and quantum dynamics, Reviews of Modern Physics 85, 1191 (2013). 13 Y Kawaguchi and M Ueda, Spinor Bose–Einstein condensates, Physics Reports 520, 253 (2012). 11

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⎛ ℏ2 ⎞ ψm* ⎜ − ∇2 ⎟ ψm, ⎝ ⎠ 2 m m =−1 m=1

T (r) =

∑

(3.42)

the total particle density m=1

n(r) =

∑

(3.43)

ψm* ψm,

m =−1

and the spin–spin interaction energy density is m, n = 1

Fα(r) =

∑

ψm* ( fα )mnψn.

(3.44)

m, n =−1

The fα are the 3 × 3 spin-1 Pauli matrices:

⎛ ⎞ 1 ⎜0 1 0⎟ fx = 1 0 1 ; 2 ⎜⎝ 0 1 0 ⎟⎠

⎛ ⎞ i ⎜0 − 1 0 ⎟ fy = 1 0 −1 ; 2 ⎜⎝ 0 1 0 ⎟⎠

⎛1 0 0 ⎞ and fz = ⎜⎜ 0 0 0 ⎟⎟ , (3.45) ⎝ 0 0 − 1⎠

and the order parameter, and the quasiparticles are described by three-component spinors:

⎛ ψ1(r) ⎞ ⎜ ⎟ ψ (r) = ⎜ ψ0(r) ⎟ ; ⎜ ⎟ ⎝ ψ−1(r)⎠

⎛ u q,1(r) ⎞ ⎟ ⎜ u q(r) = ⎜ u q,0(r) ⎟ ⎟ ⎜ ⎝ u q,−1(r)⎠

and

⎛ vq,1(r) ⎞ ⎟ ⎜ vq(r) = ⎜ vq,0(r) ⎟. ⎟ ⎜ ⎝ vq,−1(r)⎠

(3.46)

The formation of a spinor BEC is associated with at least two spontaneously broken continuous symmetries. The condensate acquires a definite phase coherence corresponding to the U(1) symmetry breaking and the atomic spins acquire a definite spin coherence corresponding to the SO(3) symmetry breaking. The broken symmetry phases form a group, such as G = U(1) × SO(3) and they are connected by the action of the group elements (rotations)

g = e iγ e−ifzαe−ifyβe−ifzα′,

(3.47)

where γ is the angle of a gauge transformation, fi are spin-F Pauli matrices and α, β and α′ are Euler angles of a 3D spin rotation. The order parameter symmetries may also be described by the group G = U(1) × SU(2), where SU(2) is a covering group for SO(3). In this case the spinor order parameters are connected via rotations

R = e iγ e−iθ n·f ,

(3.48)

where the first term on the right is a gauge transformation and the second term is a spin rotation by an angle θ about an axis determined by a 3D unit vector n , and f is a vector of spin-1/2 Pauli matrices. Depending on the details of the Hamiltonian the symmetry of the ground state may be further reduced to

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G/H =

U(1) × SU(2) , H

(3.49)

where H describes the remaining symmetries of the order parameter. In this case the values of γ , θ and n will also be restricted by the remaining symmetries of the Hamiltonian that also specifies the allowed types of topological excitations. Equation (3.48) encapsulates the special property of spinor BECs that the spin and gauge degrees of freedom are coupled. This has direct physical consequences such as the possibility of an interconversion between mass currents and spin currents.

3.9 Spin-2 Bose–Einstein condensates FRS: spin-2 phase diagram14,15,16 The Hamiltonian density for an F = 2 spinor BEC in the absence of external magnetic fields is

H = T (r) + Vext(r) +

c0 c c n(r)2 + 1 ∣F (r)∣2 + 2 ∣A(r)∣2 , 2 2 2

(3.50)

where the five terms on the right are, respectively, the kinetic energy density, external single-particle potentials, density–density contact interaction, spin–spin interaction and the spin–singlet pair amplitude density

(

A(r) = 2ψ2ψ−2 − 2ψ1ψ−1 + ψ02

)

5.

(3.51)

The effective coupling constants c0 = 4π ℏ2(4a2 + 3a 4 )/7m, c1 = 4π ℏ2(a 4 − a2 )/7m and c2 = 4π ℏ2(7a 0 − 10a2 + 3a 4 )/7m are expressed in terms of the bare s-wave scattering lengths ai corresponding to different total angular momentum channels of the colliding atoms. A ground-state phase diagram in the absence of external magnetic fields is shown in figure 3.1. In the presence of magnetic fields, the phase diagram is even richer. In particular, weak Zeeman splitting breaks the symmetry of the nematic phase order parameter into biaxial and uniaxial nematic phases with lower symmetry. The representative spinor order parameters (figure 3.2) are now five-component spinors and the spin-gauge rotations involve 5 × 5 spin-2 Pauli spin matrices fz = diag(2, 1, 0, −1, −2):

C V Ciobanu, S -K Yip and T -L Ho, Phase diagrams of F = 2 spinor Bose–Einstein condensates 61 033607 (2000). 15 Y Kawaguchi and M Ueda, Spinor Bose–Einstein condensates, Physics Reports 520, 253 (2012). 16 D M Stamper-Kurn and M Ueda, Spinor Bose gases: Symmetries, magnetism and quantum dynamics, Reviews of Modern Physics 85, 1191 (2013). 14

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Figure 3.2. Illustration of the spherical harmonics (upper row) and Majorana star (lower row) representations for a selection of F = 1 and F = 2 spinor order parameters. The integers next to the Majorana stars (red) denote the multiplicities of the complex zeros.

⎛0 ⎜ ⎜1 ⎜ ⎜ fx = ⎜ 0 ⎜ ⎜ ⎜0 ⎜ ⎝0

1 0 3 2 0 0

0 3 2 0 3 2 0

0⎞ ⎟ 0 0⎟ ⎟ 3 ⎟ 0⎟, 2 ⎟ ⎟ 0 1⎟ ⎟ 1 0⎠ 0

⎛0 − i 0 0 0⎞ ⎜ ⎟ 3 ⎜i 0 0⎟ 0 −i ⎜ ⎟ 2 ⎜ ⎟ 3 3 fy = ⎜ 0 i 0 ⎟. 0 −i ⎜ ⎟ 2 2 ⎜ ⎟ 3 ⎜0 0 0 i − i⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎝0 0 0 0⎠ i

(3.52)

A prominent and qualitatively new feature arising in spin-2 BECs is that the first homotopy group π1 of the cyclic and biaxial nematic states are non-abelian and host non-abelian vortices. This property leads to the physics of non-abelian quantum turbulence and potential applications of topological quantum information processing and storage with non-abelian vortex anyons.

3.10 High-spin Bose–Einstein condensates FRS: non-abelian phases in spinor BECs17, non-abelian vortex anyons18 Generally, a spin-F condensate is described by a 2F + 1 component spinor order parameter and correspondingly large number of angular momentum channels for 17 B Lian, T -L Ho, and H Zhai, Searching for non-Abelian phases in the Bose–Einstein condensate of dysprosium Physical Review Letters 85, 051606 (2012). 18 T Mawson, T C Petersen and T Simula, Braiding and fusion of non-Abelian vortex anyons, arXiv:1805.10009 (2018).

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two-particle interactions. Consequently, the number of possible ground-state phases and topological defects grows dramatically with increasing F. The vortex physics of scalar BECs with F = 0 is determined by the mass currents due to the curl free superfluid velocity field. The scalar BECs are qualitatively different from F = 1 spinor BECs in that the latter enable also spin currents in addition to the mass currents. The combination of the two allows fractional vortices to exist. Phenomena absent in scalar condensates include spin waves (magnon), fractional vortices, and interconversion between mass and spin currents as observed in the Einstein–de Haas effect. New physics emerges also in F = 2 BECs. Whereas all of the vortices in F ⩽ 1 are abelian anyons, the F = 2 systems support ground-state phases with vortices that are non-abelian anyons. It is possible that all systems with F ⩾ 2 may be able to host ground states characterised by non-abelian symmetry groups and the associated non-abelian vortex anyons potentially useful for topological quantum information processing and storage.

3.11 Representations of spinor Bose–Einstein condensates FRS: Majorana star and spherical harmonics representations19 Spin-F condensate spinor wave function Ψ(r) at a specific point r = r0 of space is a complex valued vector with 2F + 1 components. As an example the spinor Ψ(r0) ≡ (… , ψ1(r0), ψ0(r0), ψ−1(r0), …) could have the value of (1, 0, 0) corresponding to a ferromagnetic state of a spin-1 BEC, or (1, 0, 2 i , 0, 1)/2 corresponding to a tetrahedral cyclic state of a spin-2 BEC. The spherical harmonic representation is obtained by the mapping F

∑

d=

ψm(r0)Y Fm(θ , ϕ),

(3.53)

m =−F

where Yℓm is the spherical harmonic function. A convenient graphical representation is obtained by plotting an isosurface of ∣d ∣2 , as in figure 3.2, and using a colormap defined by the phase arg (d ). The spherical harmonics representation is particularly useful for the purpose of identifying topological excitations such as vortices. The Majorana star representation is obtained by mapping the spinor wave function onto an order 2F complex valued polynomial 2F

P (z ) =

∑ k=0

⎛ 2F ⎞ * ⎜ ⎟ ψ F − k (r)z k , ⎝ k ⎠

(3.54)

where the spinor component wave functions ψm determine the coefficients of these polynomials. For F = 1 and F = 2 these polynomials are, respectively, 19

Y Kawaguchi and M Ueda, Spinor Bose–Einstein condensates, Physics Reports 520, 253 (2012).

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Quantised Vortices

P1(z ) = ψ1*z 2 +

2 ψ0*z + ψ−*1,

P2(z ) = ψ2*z 4 + 2ψ1*z 3 +

6 ψ0*z 2 + 2ψ−*1z + ψ−*2.

(3.55)

The complex roots P (z ) = a + ib = 0 of these polynomials are mapped via a stereographic projection:

⎛ 2b 2a a 2 + b2 − 1 ⎞ , , (x , y , z ) = ⎜ 2 ⎟ ⎝ a + b2 + 1 a 2 + b2 + 1 a 2 + b2 + 1 ⎠

(3.56)

onto the unit sphere as ‘Majorana stars’—the red markers in figure 3.2.

3.12 Exotic interactions FRS: dipole-dipole interactions20, synthetic gauge fields21 A rather generic Hamiltonian for a weakly interacting BEC may be considered:

H = H0 + Vext − VPz − ΩLz + Hadd,

(3.57)

where H0 is a generic single-particle operator and the external potential Vext is used for confining and manipulating the atoms. Typical examples include harmonic or uniform traps, conventional optical lattices, vortex pinning laser potentials and in general arbitrary single-particle potentials. The next two terms, where Pz and Lz are the linear and orbital angular momentum operators, account for transformations to translating and rotating frames with speed V and angular frequency Ω, respectively. More generic frame transformations could be considered by combining rotations and translations about multiple Cartesian axes. The last term, Hadd in equation (3.57), may include more ‘exotic’ potentials and interactions, such as the following. • Dipole–dipole interactions with spin polarised condensates:

Hadd =

∫

Cdd 1 − 3 cos(θ ) n(r′)dr′ , 4π ∣r − r′∣3

(3.58)

where the constant Cdd determines the strength of the dipole–dipole interactions, θ is the angle between the polarisation direction of the spins and their relative position vector. New phenomena due to the generically long-ranged and anisotropic dipole–dipole interactions include ferrofluid behaviour, strong Bogoliubov rotons, and self-bound droplets.

20 T Lahaye, C Menotti, L Santos, M Lewenstein and T Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics 72 126401 (2009). 21 N Goldman, G Juzeliũnas, P Öhberg and I B Spielman, Light-induced gauge fields for ultracold atoms, Reports on Progress in Physics 77 126401 (2014).

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• Generic synthetic gauge fields:

Hadd = CSL s · p,

(3.59)

where CSL is a constant, p is a generic momentum operator of the particles, and the spin operator s may come in a variety of forms including an internal spin degree of freedom of the atoms, magnetic field or an external rotation. The effect of such added Hamiltonians may be discussed in terms of artificial vector potential A that changes the canonical momentum of the atoms according to p′ = p − A . New phenomena due to such engineered interactions include synthetic electromagnetic fields, spin–orbit couplings and optical flux lattices. Addition of each new term in the Hamiltonian generally changes the symmetry group of the ground-state order parameter manifold and thereby the types of topological excitations supported. The cold atom systems afford a plethora of experimental techniques to engineer and control Hamiltonians, enabling them to be deployed for quantum simulations of condensed matter systems and laboratory particle physics and cosmology.

3.13 Bardeen–Cooper–Schrieffer mean-field theory FRS: BCS theory22,23 Conventional superconductors and certain Fermi gases may be modeled within the Bardeen–Cooper–Schrieffer (BCS) mean-field theory using the Bogoliubov–de Gennes (BdG) equations

⎛ −1 ⎞ ∇2 − EF⎟ u q(r) + Δ(r)vq(r) = Equ q(r) ⎜ ⎝ 2kFξ0 ⎠

(3.60)

⎛ −1 ⎞ −⎜ ∇2 − EF⎟ vq(r) + Δ*(r)u q(r) = Eqvq(r) ⎝ 2kFξ0 ⎠

(3.61)

where kF is the Fermi wave number, ξ0 = ℏvF /Δ0 is the coherence length expressed in terms of the Fermi velocity vF, and the superconducting bulk gap function Δ0 at zero temperature, and EF = ξ0kF /2 is the Fermi energy. The pair potential

Δ(r) = 〈ψ (r)ψ (r)〉 = g

∑

u q(r)vq*(r)[1 − 2f (Eq )]

∣ Eq ∣⩽ ED

22 23

A L Fetter and J D Walecka, Quantum Theory of Many-Particle Systems, Dover (2003). J Ketterson and S Song, Superconductivity, Cambridge University Press (1999).

3-14

(3.62)

Quantised Vortices

is expressed in terms of the coupling constant g, the Debye cutoff ED, and the Fermi distribution

fq = f (Eq ) = 1/(e Eq

k BT

+ 1).

(3.63)

The normalisation conditions for the quasiparticle amplitudes are

∫ {u q*(r)up(r) + vq*(r)vp(r)}d r = δqp

(3.64)

and the entropy can be expressed in terms of the fermionic quasiparticles as

SF = −kB ∑{(1 − fq ) ln(1 − fq ) + fq ln( fq )}.

(3.65)

q

For Fermi systems there is no macroscopic wave function and an equivalent of GP equation is absent. Consequently, the equality between the single (quasi)particle excitations and density fluctuations is not present in the Fermi system and the macroscopic dynamics of the system involves many quasiparticles. The system can be solved for a uniform system with a plane wave ansatz resulting in the excitation spectrum

⎛ p2 ⎞2 q 2 ⎜ ⎟ ⎜ 2m − EF⎟ + Δ0 , ⎝ ⎠

Eq =

(3.66)

which has the bulk energy gap Δ0 . The gap and the critical temperature are related by

kBTc ∝ Δ0 ∝ EFe−1 N (EF )g ,

(3.67)

where N (EF ) is the density of states per unit volume. The variational many-body BCS ground state

ΨBCS =

∏ (uk + vk αk†↑α−†k↓)∣vac〉

(3.68)

k

with quasiparticle normalisation u k2 + vk2 = 1 describes a sea of Cooper-pairs excited out of the vacuum ∣vac〉.

3.14 Ultracold Fermi gases FRS: theory of ultracold atomic Fermi gases24

24

S Giorgini, L P Pitaevskii, and S Stringari Theory of ultracold atomic Fermi gases, Reviews of Modern Physics 80, 1215 (2008).

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Quantised Vortices

For a homogeneous spin polarised Fermi gas of N atoms the density n and Fermi wave number kF are related by

kF3(r) 6π 2

(3.69)

ℏ2kF2(r) . 2m

(3.70)

n(r) = and the local Fermi energy is given by

EF =

Within a local density approximation the density of a trapped gas is thus

n(r) =

⎤3 2 1 ⎡ 2m ( μ − V ( r )) ⎢ ⎥⎦ . 6π 2 ⎣ ℏ2

(3.71)

For an isotropic 3D harmonic trapping, V (r) = mωr 2 /2, the chemical potential μ = kBTF = (6N )1/3ℏω, with T the Fermi temperature. Below the superfluid transition temperature, such Fermi gas features a BEC-BCS cross over. The scattering length a, is tuneable via Feshbach resonance

⎛ ΔB ⎞ a = abg ⎜1 − ⎟, ⎝ B − B0 ⎠

(3.72)

where a bg is the background scattering length and ΔB is the width of the resonance centered at magnetic field B0. Three limiting behaviours may be identified as a is varied. At unitary limit, 1/kFa = 0, the fermions are strongly interacting and all length scales associated with the particle interactions disappear and are replaced by universal behaviour. In the BCS limit, 1/kFa → −∞ the fermions are loosely bound forming Cooper pair molecules. The system behaviour is then described by the fermionic BdG theory with MFermi and closely follows the theory of BCS superconductors. In the BEC limit, 1/kFa → ∞ the fermions are tightly bound into dimers that are effectively described as bosons. The system behaviour is then described by bosonic BdG theory with MBose and closely follows the theory of BECs. In the superfluid phase the macroscopic behaviour can still be modelled using the quantum hydrodynamical equations with a simple modification to the equation of state. However, the microscopic details are strongly affected by the details of the particle interactions. Quantised vortices may exist on both sides of the BEC–BCS cross over but the internal structure of the vortices is different. On BEC side, at zero temperature, the vortex cores are practically empty despite hosting kelvon quasiparticles. On the BCS side the vortex core is filled with Caroli–de Gennes–Matricon quasiparticle modes of many different momenta. The combination of Fermi statistics, spin and particle interactions yield a richness of phenomena in ultracold Fermi gases. For example, spin imbalance may give rise to inhomogeneous Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states and chiral p-wave pairing interactions in 2D systems are predicted to yield

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Quantised Vortices

Majorana zero modes within the vortex cores topologically protected by zero energy edge modes.

3.15 Dirac–Bogoliubov–de Gennes systems FRS: Dirac-BdG systems25 Certain 2D condensed matter systems, including bilayer graphene, have electronhole excitation spectra that may be calculated using the Bogoliubov–de Gennes matrix of type MFermi . This may result in a massless Dirac–BdG system, E(r) = 0, with a traceless BdG matrix

MFermi = ℏvF q · σ ,

(3.73)

where vF is the Fermi velocity, q = (qx , qy, 0) is a 2D momentum vector, and σ is a vector of spin-1/2 Pauli matrices. A massive case is realized with

MFermi = ℏvF q · σ + mσz .

(3.74)

The Dirac–BdG equations have a structure whose properties can broadly be characterized in terms of generic features of their excitation energy spectra as illustrated in figure 3.3.

3.16 Gapless, massless, linear spectra FRS: gapless excitations26 Figure 3.3(a) illustrates a typical gapless excitation spectrum with a linear dispersion relation

Eq = pq cs ,

(3.75)

where the constant cs is the speed of propagation of the quasiparticle. In a BEC, the long-wavelength Bogoliubov phonons have energy which is a linear function of momentum and this spectral feature results in a non-zero Landau critical velocity and may be viewed to be responsible for superfluidity in such systems. The photon in free space is massless and its energy is a linear function of its momentum. In many condensed matter electronic ‘Dirac materials’, such as a bilayer graphene, the effective mass of the electrons vanish at the Dirac points, in the vicinity of which their dispersion relation is linear and is frequently referred to as a Dirac cone due to the similarity with the dispersion relation for massless fermionic particles described by the relativistic Dirac equation. Depending on the quantum statistics of the

25 26

M Z Hasan and C L Kane, Colloquium: Topological insulators, Reviews of Modern Physics 82, 3045 (2010). P C Hohenberg and P C Martin, Microscopic theory of superfluid helium, Annals of Physics 34, 291 (1965).

3-17

Quantised Vortices

Figure 3.3. Spectral features of generic quasiparticle systems.

system, the (doubled) zero mode depicted with red and blue markers in figure 3.3(a) may be either a Nambu–Goldstone boson or a Majorana fermion zero mode.

3.17 Gapped, massive, quadratic spectra FRS: gapped excitations24 Figures 3.3(b) and (c) illustrate typical excitation spectra with the quadratic dispersion relation pq2 (3.76) , Eq = Δ + 2m* where m* is an effective mass. The spectrum in (b) is gapless in the sense that Δ = 0. The spectrum in (c) is gapped in the sense that Δ > 0. There are no quasiparticle states within this energy gap and therefore a minimum energy of Δ must be supplied in order to create an excitation in such a system. In a usual band insulator the energy gap is so large ≈ 101 eV that effectively no conduction of electrons is possible for practical values of applied voltages. In a semiconductor the energy gap is ≈ 100 eV and therefore reasonable bias voltages are sufficient to close the gap to achieve charge conduction. In a BCS superconductor the energy gap is very small ≈ 10−3 eV, yet crucial for the phenomenon of superconductivity as the absence of scattering states at the Fermi surface EF is associated with a critical current and superconducting behaviour. Figure 3.3(d) illustrates a dispersion relation for a system which has an effective energy gap and is an insulator in the bulk but has gapless metallic (conducting) surface states.

3-18

IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 4 Topological invariants and quantities

4.1 Topology and ordered structures FRS: topological defects1,2 Topological structures in ordered media may be classified by topological invariants— numbers which are unaffected by local perturbations to the system and only depend on the topology of the space in which the topological structure exists. Typically, changing the system’s topology costs a large fraction of the total energy of the system, which makes topological states particularly robust. In addition to topological ‘defects’, such as vortices, that exist in materials, whole physical systems may also be topologically ordered. Notable examples of the latter include quantum Hall states of two-dimensional (2D) electron gas and topological insulators. Phase transitions between different topological states may occur even at zero temperature, in which case they are known as quantum phase transitions and are mediated by quantum fluctuations. Topological invariants and quantities are ubiquitous in physics and mathematics and the discussion is here limited to a few that are frequently encountered in the context of superfluid systems.

4.2 A game of lines and loops FRS: dislocations3

1

N D Mermin, The topological theory of defects in ordered media, Reviews of Modern Physics 51, 591 (1979). M Nakahara, Geometry, Topology and Physics, CRC Press (2003). 3 J P Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Oxford University Press (2006). 2

doi:10.1088/2053-2571/aafb9dch4

4-1

ª Morgan & Claypool Publishers 2019

Quantised Vortices

Figure 4.1. A game of lines and loops. The integer counts the net winding number of topological defects enclosed by the loop.

Start by drawing multiple continuous non-crossing lines (curves) on a piece of paper. Physically, these could represent, for example, wave fronts of propagating light or rows of atoms in a crystal. Next, draw an arbitrary closed loop on the paper and choose one point on this loop as a base point to start from. Then fix a 2D Cartesian (x, y) coordinate system and ‘walk’ around the loop. The game has two simple rules: (i) every time you cross a line in the positive x-direction you add one to a counter shown in the corner of each subfigure 4.1, and (ii) every time you cross a line in the negative x-direction you subtract one from the counter. When you return to the starting point, the counter will show an integer and this number is a topological invariant that counts the net winding number of topological defects enclosed by the loop. Figure 4.1 shows a few realisations of this game. It is clear that the absolute sign of the obtained number depends on which direction one travels around the loop. Therefore the absolute sign of this invariant is a subjective observable whereas the relative sign between two loops traveled in specific directions is significant. In this example, the set of lines define an order parameter space and the numbers calculated are the topological invariants of this order parameter space. This order parameter space could correspond to many different physical systems, all of which would have the same underlying topological structure. For example, the discretised positions of atoms in a 2D crystal could be joined and mapped onto continuous lines. The nonzero values of the counters would then be identified with screw dislocations of the crystal.

4.3 Maps and order parameters FRS: defects in ordered media4

4

N D Mermin, The topological theory of defects in ordered media, Reviews of Modern Physics 51, 591 (1979).

4-2

Quantised Vortices

Figure 4.2. Mapping the physical space to an order parameter space. The blue arrow represents a vortex filament in physical space. Each point along the path encircling the vortex is mapped onto a complex number (vector in the Argand plane). This vector rotates an integer number of revolutions under this mapping and this integer is a topological invariant associated with the vortex.

To classify the topological structures and phases of a physical system, the properties of the physical system are first mapped onto an order parameter space, as illustrated in figure 4.2. The allowed symmetries of the order parameter space then determine the types of topological structures that may exist in the physical system. The topologically allowed objects are classified by the homotopy groups characterising the order parameter space. Whether the topologically allowed objects are actually realised in any particular physical system depends on energetic and dynamical considerations.

4.4 Homotopy classification of defects FRS: homotopy classification of defects5,6

Generically, the homotopy groups πn(M , x0 ), where M is the order parameter manifold and x0 is the base point, classify the allowed topological defects in the system. Here

n=d−m−1

(4.1)

is the dimension of the sphere Sn enclosing the m-dimensional topological defect embedded in a d-dimensional physical space. Table 4.1 lists a few symmetry groups together with their first, second, third and fourth homotopy classes. In a threedimensional, d = 3, physical space π1 classifies vortices and π2 classifies monopoles. In a two-dimensional, d = 2, physical space vortices appear as point-like objects and

5

L Michel, Symmetry defects and broken symmetry. Configurations Hidden Symmetry, Reviews of Modern Physics 52, 617 (1980). 6 Y Kawaguchi and M Ueda, Spinor Bose–Einstein condensates, Physics Reports, 520, 253 (2012).

4-3

Quantised Vortices

Table 4.1. Homotopy classification of topological defects associated with a number of symmetry groups.

Symmetry group

π1

π2

π3

π4

Relevant system

U(1) SO(3) SU(2)  P2  P3 SU(2)/ S2× U(1)/ 2 SO(3)×U(1)/ S2× SO(3)/ 2 S1× SO(3) S1× S2

 2 0 2 2    4  × 2 

0 0 0  0 0  0  0 

0     0   ×  

0 2 2 2 2 2 2 2 2 × 2 2 2

scalar BEC He3-A (dipole-locked) electron spin nematics ferromagnetic BEC biaxial nematics polar F = 1 BEC cyclic F = 2 BEC He3-A (dipole-free) He3-B (dipole-free) He3-B (dipole-locked)

hence π1 classifies both vortices and monopoles. A zero in the table means that a topologically stable defect of that kind is not allowed.

4.5 Burgers vector FRS: Burgers vector7 Consider a 2D imperfect crystal lattice of atoms, shown in figure 4.3. One may then draw a continuous loop (p-sided polygon) in the lattice by joining the neighbouring p (orange) atoms along the path with p vectors. The resultant vector R = ∑i =1 pi is known as the Burgers vector, which when normalised with respect to a corresponding perfect lattice, counts the total ‘charge’ of the enclosed defects. One may also apply the rules of the game of lines and loops to this atomic structure by replacing the discrete atoms with continuous lines to obtain an integer that corresponds to the magnitude of the normalised Burgers vector, which in this case is also the winding number of this topological defect.

4.6 Gauss–Bonnet theorem FRS: Gauss-Bonnet theorem8 The Gauss–Bonnet (GB) theorem 7 J M Burgers, On the emergence of patterns of order, Bulletin of the American Mathematical Society 69, 1 (1963). 8 A F Kharshiladze et al In: Encyclopaedia of Mathematics: Coproduct–Hausdorff–Young Inequalities, Springer (1995).

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Quantised Vortices

Figure 4.3. Topological dislocation defect in a two-dimensional (2D) crystal lattice quantified by the Burgers vector.

ω + τ = 2πχ (M)

(4.2)

states that the sum of the total curvature

ω=

∫M KdS

(4.3)

of a compact 2D Riemannian manifold M, and the rotation

τ=

∫∂M kgdl

(4.4)

of its smooth surface ∂M are equal to 2π times the Euler characteristic χ(M) of M. Here K is the Gaussian curvature, dS is an area element, kg is geodesic curvature, and dl is a line element of the boundary. For orientable compact surfaces without boundary χ (M ) = (2 − 2g ), where g is the genus (the number of holes) of the surface, the GB theorem reduces (τ = 0) to

ω = 2π (2 − 2g )

(4.5)

providing a direct link between the geometrical and topological properties of the 2π

π

system. For example, for a smooth sphere g = 0 and ω = ∫ ∫ (1/R2 )R2dθdϕ = 4π , 0 0 and for a torus (coffee cup with a handle) g = 1 and ω = 0 as the ‘inner’ and ‘outer’ curvatures of the torus have opposite signs and cancel out.

4.7 Winding number FRS: winding number9

9

W Fulton, The Winding Number. In: Algebraic Topology. Graduate Texts in Mathematics, vol 153, Springer (1995).

4-5

Quantised Vortices

Consider a simple superfluid such as a Bose–Einstein condensate (BEC) or heliumII described by U(1) symmetry. First draw a closed loop S1 within the superfluid, as shown in figure 4.2(left). Each point in the fluid along this loop can be mapped onto a complex number in the order parameter space figure 4.2(right). As the loop in the physical space is traversed once, the phasor in the order parameter space may wind around the origin an arbitrary (integer) number of times. This winding number w is a topological invariant associated with the (broken) symmetry of the order parameter space and it characterises the topological excitations of the system, which in this case are called quantised vortices, shown in figure 4.2(left) with a blue oriented curve. A vortex has a positive integer winding number w > 0 whereas a vortex with a negative winding number w < 0 is called an antivortex. A vortex and an antivortex of equal strengths can be annihilated resulting in a topologically trivial (vacuum) state. The reverse process whereby the vacuum spawns a vortex–antivortex pair is called a pair creation. In general, a combination (fusion) of two vortices with winding numbers w1 = n and w2 = m yields a new vortex with winding number n + m.

4.8 Berry phase, curvature, and connection FRS: geometric phases10,11,12,13,14,15 An adiabatic evolution of a quantum state ψ (r), accumulates a geometric phase

γ (C ) = i

∫C 〈ψ ∣∇R ψ 〉 · d R

(4.6)

as it is transported along a path C determined by the set of parameters R. By defining a Berry connection

A(R) = i〈ψ (R)∣∇R ∣ψ (R)〉

(4.7)

Ω(R) = ∇R × A(R),

(4.8)

and Berry curvature

the Berry phase may be expressed as 10 S Pancharatnam, Generalized theory of interference, and its applications, Proceedings of the Indian Academy of Sciences 44, 247 (1956). 11 M V Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society A 392, 1802 (1984). 12 F Wilczek and A Zee, Appearance of gauge structure in simple dynamical systems, Physical Review Letters 52, 2111 (1984). 13 R Simon and N Mukunda, Bargmann invariant and the geometry of the Güoy effect, Physical Review Letters 70, 880 (1993). 14 J Anandan, J Christian and K Wanelik, Resource Letter GPP-1: Geometric Phases in Physics, American Journal of Physics 65, 180 (1997). 15 F Wilczek and A Shapere, Eds, Geometric Phases in Physics, Advanced Series in Mathematical Physics 5, 528 (1989).

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Quantised Vortices

γ=

∮C A(R) · d R = ∫A Ω(R) · dA,

(4.9)

where the last two forms are related via Stokes theorem. The Berry phase may also be expressed in an operationally (experimentally and computationally) convenient, discretised form (with label i discretising the path C) as

⎛ P ⎞ γ (C ) = −Im ⎜⎜ln ∏ 〈ψi∣ψi +1〉⎟⎟ , ⎝ ⎠ i

(4.10)

which is essentially a negative of the Bargmann invariant. The Berry phase γ (C ) may be extended by considering adiabatic evolution of systems with degenerate state spaces, in which case γ (C ) becomes replaced by a unitary matrix Γ(C ), generally referred to as a non-abelian Berry phase or the Wilczek–Zee phase.

4.9 Chern number FRS: Mermin-Ho relation16, TKNN number17, Chern number18, topological insulators19,20 Whereas the winding number, which counts how many multiples of 2π the phase of the order parameter wraps upon encircling a topological defect, is a useful topological invariant for classifying line-like objects such as quantised vortices, the Chern number is a topological invariant that counts how many multiples of solid angle 4π (coverings of the Bloch sphere) the generalised vorticity of the order parameter covers the unit sphere upon enclosing a topological defect, and is useful for, but not restricted to, classifying point-like objects such as monopoles and topological phases of topologically ordered materials. Consider, for instance, the order parameter for 3He or ferromagnetic spin-1 BECs, which may be expressed as

ψ (r) =

ρs (m + i n)

(4.11)

in terms of the normalised local spin vector l = 〈F〉/ρs with the spin texture l = m × n , such that l 2 = 1. The superfluid velocity is

16 N D Mermin and T -L Ho, Circulation and Angular Momentum in the A Phase of Superfluid Helium-3, Physical Review Letters 36, 594 (1976). 17 D J Thouless, M Kohmoto M P Nightingale and M den Nijs, Quantized Hall Conductance in a TwoDimensional Periodic Potential, Physical Review Letters 49, 405 (1982). 18 T Fösel, V Peano and F Marquardt, L lines, C points and Chern numbers: understanding band structure topology using polarization fields, New Journal of Physics 19, 115013 (2017). 19 X -L Qi and Z -C Zhang, Topological insulators and superconductors, Reviews of Modern Physics 83, 1057 (2011). 20 M Z Hasan and C L Kane, Colloquium: Topological insulators, Reviews of Modern Physics 82, 3045 (2010).

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Quantised Vortices

vs =

ℏ mi ∇ni 2m

(4.12)

and the generalised vorticity is determined by the Mermin–Ho relation

∇ × vs =

ℏ ϵαβγ lα∇lβ × ∇l γ. 2m

(4.13)

The line integral

2m ℏ

∮∂S vs · dℓ = 4πC

(4.14)

over any closed path ∂S along which the l -vector texture points in the same direction is an invariant with the Chern number

C=

1 4π

∬S (∇ × vs ) · eˆ zdS

(4.15)

being related to equation (4.14) via the Stokes theorem. Substituting the Mermin– Ho relation yields

C=

ℏ 8mπ

⎛

⎞

∬S l · ⎜⎝ ∂∂xl × ∂∂yl ⎟⎠dxdy.

(4.16)

The Chern number is also useful for classifying topological phases of matter such as topological insulators. In the (pseudo)spin-1/2 case it may be expressed as

C=

1 2π

∬BZ (∇k × A n(k)) · kˆ zdkxdky,

(4.17)

where An(k) = i〈un(k)∣∇k ∣un(k)〉 is the Berry connection whose vorticity projection along the eˆ z axis over the Brillouin zone is evaluated in terms of the Bloch wave functions un(k) where k is the quasimomentum and n is the band index. With the aid of the Mermin–Ho relation, equation (4.13), this yields the TKNN number

Cn =

1 2π

⎛

∂u n(k) ⎝ ∂ky

∬BZ ⎜⎜

∂u n(k) ∂kx

−

∂u n(k) ∂kx

⎞ ∂u n(k) ⎟ ⎟dkxdky ∂ky ⎠

(4.18)

of the band n.

4.10 Linking number, writhe and twist FRS: linking number, writhe and twist21

21 M A Berger, L H Kauffman, B Khesin, H K Moffatt, R L Ricca, D W Sumners, Lectures on Topological Fluid Mechanics, Springer-Verlag (2009).

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Quantised Vortices

Let s1(ξ ) and s 2(ξ ) be the position vectors of two space curves K1 and K2 parametrised by the variable ξ with d s(ξ )/dξ denoting the unit tangent vector. The Gauss linking number (integer) of the two oriented disjoint loops is

Lk(K1, K2) =

1 4π

⎡

∮K ∮K ⎢⎣ ddsξ1 × ddsξ2′ 1

2

·

s1(ξ ) − s 2(ξ′) ⎤ ⎥dξ dξ′ ∣s1(ξ ) − s 2(ξ′)∣3 ⎦

(4.19)

and it equals one half of the total number of oriented crossings of any planar projection of the two linked loops. The linking number can be decomposed in terms of the writhe Wr and twist Tw as

Lk(K1, K2) = Wr(K) + Tw(K)

(4.20)

where

Wr(K) = Lk(K , K) =

1 4π

⎡

⎤

∮K ∮K ⎢⎣ ddξs × ddξs′ · ∣ss((ξξ)) −− ss((ξξ′′))∣3 ⎥⎦ dξ dξ′

(4.21)

and

Tw(K) =

1 2π

⎤

⎡

∮K ⎢⎣ ddξs · n(ξ) × nd(ξξ) ⎥⎦ dξ.

(4.22)

The unit vector n(ξ ) normal to the tangent vector d s(ξ )/dξ satisfies

n(ξ ) ·

ds =0 dξ

and

s 2(ξ ) = s1(ξ ) + ϵ(ξ )n(ξ ),

(4.23)

where ϵ is small. Intuitively, the linking number characterises how many times two curves wind around each other, the writhe characterises the amount of coiling of a knot and the twist characterises the rate of rotation of a smooth ribbon around the space curve.

4.11 Helicity FRS: helicity22 Hydrodynamic helicity within volume V can be expressed in terms of the velocity v and vorticity ω = ∇ × v as

H=

∫V v · (∇ × v)d V.

(4.24)

For a single closed vortex of strength κ its helicity

H = κ 2(Wr + Tw) = κ 2Lk 22

H K Moffat, Helicity and singular structures in fluid dynamics, PNAS 111, 3663 (2014).

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(4.25)

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is invariant and directly proportional to the Gauss linking number Lk. For two linked but untwisted and unknotted vortex lines with circulations κ1 and κ2 the helicity

H = ±2κ1κ2Lk.

(4.26)

4.12 Enstrophy FRS: enstrophy23 An enstrophy is the integral of squared vorticity

Ω=

∫A ∣ω∣2 dA = ∫A ∣∇ × v∣2 dA

(4.27)

and it is a conserved quantity in 2D Eulerian incompressible hydrodynamics. For superfluids for which ω = κ ∑v δ (r − rv), the enstrophy effectively counts the total number of vortices in the system, irrespective of their sign of circulation. Hence in a simple BEC the enstrophy (divided by κ 2 ) is equal to the genus of the condensate wave function and counts the order of the multiplicity of its topological connectedness.

4.13 Kauffman bracket polynomial FRS: bracket polynomial24 The Kauffman bracket polynomial is useful for classifying knots and links. Calculation of the Kauffman bracket 〈K〉 of a knot K follows three simple rules, shown pictorially in figure 4.4. The Kauffman bracket of a trivial link O equals identity, 〈O〉 = 1 (first row). Disjoint union of loops results in a multiplication of the remaining knot by a polynomial factor 〈O ∪ L〉 = ( −A2 − A−2 )〈L〉 (second row), and the skein relations double the number of operations in the calculation of the polynomial as illustrated in the two last rows of the figure.

4.14 Jones polynomial FRS: Jones polynomial25, Hopf link26, knot atlas27

23

R H Kraichnan and D Montgomery, Two-dimensional turbulence, Reports on Progress in Physics 43 547 (1980). 24 L H Kauffman, State models and the Jones polynomial, Topology 26, 395 (1987). 25 V F R Jones, A polynomial invariant for knots via Von Neumann algebras, Bulletin of the American Mathematical Society 12, 103 (1985). 26 Heinz Hopf, Collected works, Springer (2013). 27 Knot Atlas (http://katlas.org/wiki/Main_Page).

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Figure 4.4. Set of diagrammatic rules for calculating the Kauffman bracket polynomial of knots and links, and the writhe of oriented crossings.

The Jones polynomial of a knot or a link K

JK(t ) = ( −A3)−Wr(K)〈K〉

(4.28)

is a powerful knot invariant that may be obtained via substitution t = A−4 after multiplying the Kauffman bracket polynomial 〈K〉 by a factor of ( −A3)−Wr(K). The writhe Wr(K) is the number of signed crossings in the knot. The sign of a crossing is determined by first choosing an orientation for all disjoint links. An ‘arrow of time’ is drawn through the crossing in the direction determined by the two paths at the crossing as shown in the last column of figure 4.4. The crossing has a positive sign if the strand on the top crosses over to the right of the arrow of time and negative if it crosses to the left. The Jones polynomial J of the twisted unknot U shown in the top row of figure 4.5 is JU (t ) = ( −A3)〈U〉 = 1 and JU (t ) = ( −A−3)〈U〉 = 1 for the twisted unknot on the second row of figure 4.5. The number of links in the Hopf link H is 2 and the writhe Wr(H) = ±2. The Kauffman bracket polynomial is calculated in figure 4.5 and equals

〈H〉 = −A4 − A−4 .

(4.29)

This illustrates how unraveling each crossing in the knot doubles up the number of computational operations required in unraveling the knot. The Jones polynomial JH+(t ) of the positive Hopf link in figure 4.5 with Wr(H+) = 2 is therefore JH +(t ) = ( − A3)−Wr(H +)〈H+〉 = A−6 ( − A4 − A−4 ) = − A−2 − A−10 = − t1 2 − t 5 2 (4.30)

where t = A−4 . Similar calculation for the Jones polynomial JH−(t ) of the negative Hopf link with Wr(H−) = −2 yields

JH−(t ) = ( −A3)−Wr(H−)〈H−〉 = A6 ( −A4 − A−4 ) = −A2 − A10 = −t −1 2 − t −5 2. (4.31) The trefoil knot T is chiral and has three links and writhe Wr = ±3. There are two distinct trefoil knots, corresponding to the two values of chirality, that cannot be

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Figure 4.5. Calculation of the Kauffman bracket polynomial for two twisted unknots and the Hopf link.

deformed to each other using a continuous transformation. The calculation of the Kauffman bracket of a trefoil knot involves 23 = 8 operations and yields

〈T ±〉 = A∓7 − A∓7 − A±5.

(4.32)

The Jones polynomial of the positive trefoil with Wr(T +) = 3 is therefore

JT +(t ) = ( −A3)−Wr(T +)〈T +〉 = A4 + A12 − A16 = t1 + t 3 − t 4.

(4.33)

The Jones polynomial of the negative trefoil with Wr(T −) = −3 is

JT −(t ) = ( −A3)−Wr(T −)〈T −〉 = A−4 + A−12 − A−16 = t −1 + t −3 − t −4.

4-12

(4.34)

IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 5 Topological excitations

5.1 Topological defects FRS: topology and physics1,2 Topological excitations are a class of particle-like entities often present in ordered media. They may be frozen in the system at the time the ordered phase was formed, or created later by supplying a considerable amount of energy, with respect to the total energy of the system. Topological excitations puncture the order parameter of the system, changing its topology. The terms ‘topological defect’ and ‘topological excitation’ are used interchangeably depending on whether one wishes to emphasise that the presence of topological defects typically make the host materials structurally weaker, or that it is the topological excitations that make the host materials physical properties richer. Topological excitations in ordered media are as ubiquitous as the zoo of elementary particles in our world.

5.2 Soliton FRS: solitary waves3, solitons4 Topological solitons may be point-, line- and plane-like objects depending on the dimensionality of the embedding space. However, the term soliton is often used 1

M Nakahara, Geometry, Topology and Physics, CRC Press (2003). J P Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Oxford University Press (2006). 3 J S Russell, Report of the 14th Meeting of the British Association for the Advancement of Science, Report on Waves 311 (1844). 4 N J Zabusky, and M A Porter, Scholarpedia, Soliton 5, 2068 (2010). 2

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more broadly. The archetypal soliton is the solitary wave, characterised by a localised over-density of a fluid, which preserves its shape as it propagates and can easily be observed in shallow water channels. Such solitons, for which nonlinear and dispersive effects on the traveling wave are perfectly balanced, can often be obtained as solutions to many nonlinear wave equations such as the Korteweg–de Vries and Gross–Pitaevskii (GP) equations. Solitons possess many particle-like properties that became particularly clear when two or more solitons collide. In cold atom gas systems so-called bright and dark solitons are of particular interest.

5.3 Bright soliton FRS: bright soliton5, bright soliton train6 In an attractively interacting, g < 0, quasi-one-dimensional (1D) Bose–Einstein condensate (BEC), a bright soliton appears as a localised over-density in the superfluid. Such bright solitons are close counterparts to classical shallow water solitary waves. The GP equation admits a bright soliton solution:

ψ (x , t ) =

⎛ x ⎞⎤−1 2μ ⎡ ⎢cosh ⎜ ⎟⎥ e−iμt ℏ, g ⎢⎣ ⎝ 2 ξ ⎠⎥⎦

(5.1)

where μ < 0 is the chemical potential and ξ is the healing length. The bright solitons do not involve a phase jump across the soliton and are not stabilised or protected topologically. Instead, they persist due to energetic considerations. The speed of propagation of a bright soliton is not determined by the shape of the soliton.

5.4 Grey and dark soliton FRS: dark solitons7,8 In a repulsively interacting, g > 0, quasi-1D BEC, a grey soliton appears as a localised under-density in the superfluid. If the density vanishes entirely at one point

5 L Khaykovich, F Schreck, G Ferrari, T Bourdel, J Cubizolles, L D Carr, Y Castin, and C Salomon, Formation of a Matter-Wave Bright Soliton, Science 296, 1290 (2002). 6 K E Strecker, G B Partridge, A G Truscott, and R G Hulet, Formation and propagation of matter-wave soliton trains, Nature 417, 150 (2002). 7 S Burger, K Bongs, S Dettmer, W Ertmer, K Sengstock, A Sanpera, G V Shlyapnikov, and M Lewenstein, Dark Solitons in Bose–Einstein Condensates, Physical Review Letters 83, 5198 (1999). 8 J Denschlag, J E Simsarian, D L Feder, C W Clark, L A Collins, J Cubizolles, L Deng, E W Hagley, K Helmerson, W P Reinhardt, S L Rolston, B I Schneider, W D Phillips, Generating Solitons by Phase Engineering of a Bose–Einstein Condensate, Science 287, 97 (2000).

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the grey soliton is called a dark soliton. The grey soliton wave function in a uniform superfluid with particle density n0 may be expressed as

ψ (x , t ) =

⎡ v n 0 ⎢i + ⎢⎣ cs

⎛ ⎛ x − vt ⎞⎤ v2 ⎞ ⎜1 − 2 ⎟ tanh ⎜ ⎟⎥e−iμt ℏ cs ⎠ ⎝ ⎝ 2 ξv ⎠⎥⎦

(5.2)

with the probability density

⎡ v2 ⎛ ⎛ x − vt ⎞⎤ v2 ⎞ n(x , t ) = n 0⎢ 2 + ⎜1 − 2 ⎟ tanh2 ⎜ ⎟⎥. ⎢⎣ cs cs ⎠ ⎝ ⎝ 2 ξv ⎠⎥⎦

(5.3)

The density, n min , in the soliton core and its velocity v are related by

n min = n 0

v2 , cs2

(5.4)

which shows that the density in the core vanishes for a stationary soliton, while as the density perturbation becomes smaller the speed of the soliton approaches the speed of sound, cs. The velocity-dependent width of the soliton core is

ξ (v ) =

ξ(0) 1−

v2

.

(5.5)

c s2

5.5 Solitonic vortex FRS: solitonic vortex9,10,11 A solitonic vortex is a vortex in a system where one spatial extent of the system in the direction orthogonal to the circulation vector is short and of the order of the width of the vortex core. From a topological point of view a solitonic vortex is equivalent to a vortex. However, due to its quasi-1D confinement the energetic behaviour of such a ‘squeezed’ vortex means that its behaviour lies somewhere between a soliton of a 1D system and a vortex of a 2D system.

9 J Brand and W P Reinhardt, Solitonic vortices and the fundamental modes of the ‘snake instability’: Possibility of observation in the gaseous Bose–Einstein condensate, Physical Review A 65, 043612 (2002). 10 M J H Ku, W Ji, B Mukherjee, E Guardado-Sanchez, L W Cheuk, T Yefsah, and M W Zwierlein, Motion of a Solitonic Vortex in the BEC–BCS Crossover, Physical Review Letters 113, 065301 (2014). 11 S Donadello, S Serafini, M Tylutki, L P Pitaevskii, F Dalfovo, G Lamporesi, and G Ferrari, Observation of Solitonic Vortices in Bose-Einstein Condensates, Physical Review Letters 113, 065302 (2014).

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5.6 Plain vortex FRS: quantised vortex12,13,14,15 A complex-valued function such as the condensate wave function of a scalar BEC may be expressed, without a loss of generality, in the Madelung form:

ψ (r) = ∣ψ (r)∣e iS(r),

(5.6)

where the real function S (r) denotes the phase of the condensate. The definition of the probability current

j=−

iℏ * (ψ (r)∇ψ (r) − ψ (r)∇ψ *(r)) = n(r)vs (r) 2m

(5.7)

allows an identification of the superfluid velocity vs(r) as the gradient of the phase

vs (r) =

ℏ ∇S (r). m

(5.8)

As such, the superflow is potential flow and therefore it is curl-free provided the phase S (r) is a smooth analytic function. In contrast, if the wave function has Nv phase singularities (vortices) the vorticity Nv

ω(r) = ∇ × vs (r) =

∑δ(r − ri )κi ,

(5.9)

i=1

where κi is a vector with magnitude ∣κi ∣ = κ = h /m, has non-zero value at the points that correspond to the locations of the vortex cores. Hence, applying the Stokes theorem shows that the circulation of a vortex is quantised:

∫A (∇ × vs ) · d a = ∮Γ vs · dℓ = wκ,

(5.10)

where the integer w is the winding number of the vortex and A denotes the area enclosed by the path Γ. This means that on traversing the singularity along a closed loop, the complex phase S (r) ‘winds’ an integer w multiple of 2π. The winding numbers are additive, w1+2 = w1 + w2 , with vortices with w > 0 called vortices and those with w < 1 antivortices (or vice versa). Vortices with ∣w∣ > 1 are multiply quantised vortices.

12

L Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6, 279 (1949). L P Pitaevskii, Vortex Lines in an Imperfect Bose Gas, Soviet Physics JETP 13, 451 (1961). 14 E P Gross, Structure of a quantized vortex in boson systems, Il Nuovo Cimento 20, 454 (1961). 15 A L Fetter, Rotating trapped Bose-Einstein condensates, Reviews of Modern Physics 81, 647 (2009). 13

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A trial wave function for a static axisymmetric vortex can be expressed as

ψ (x , y ) =

re iwθ r 2 + wrc2

n0 ,

(5.11)

where n0 is a constant particle density far from the vortex core of size w rc . The superfluid velocity of such a vortex is ℏw (5.12) eˆ z. vs (r) = mr For r ≪ rc , the condensate density n(r ) = ∣ψ (x , y )∣2 = r 2n 0 /wrc2 and for r ≫ rc it is constant, n(r ) = n 0. Hence the current density of such a vortex is well defined everywhere and grows linearly, j (r ) = ℏrn 0 /mrc2 for r ≪ rc, and decreases as an inverse power law, j (r ) = wℏn 0 /mr for r > rc .

5.7 Polynomial vortex FRS: vortices and polynomials16,17 Polynomial vortices are a mathematically convenient tool for describing the singular structure of quantised vortices. A complex function with a simple zero (vortex phase singularity) at the origin may be expressed as

ψ = re iθ = r[cos(θ ) + i sin(θ )] = x + iy = z.

(5.13)

However, this function is not L2 normalisable and therefore is not a valid quantum mechanical wave function for a vortex state. Instead, it only describes the structure of the vortex in the vicinity of the vortex core. A vortex z and an antivortex z * = x − iy are related by time-reversal (complex conjugation) that changes the direction of the superflow around the vortex core. A vortex core located at a position rv = (xv, yv ) may be expressed as

ψ (r) = z − zv = (x − xv ) + i (y − yv ).

(5.14)

A vortex phase singularity of order w may be expressed as ψ (r) = (z ± zv )w . A general node structure consisting of Nv = N+ + N− phase singularities with N+ vortices and N− antivortices may be expressed as N+

ψ (r) =

N−

∏ (z − zi ) m ∏ (z* − zi*)n , i

i=1

i

(5.15)

i=1

where the integers mi and ni are the winding numbers of each of the vortex phase singularities. Polynomial vortices are a convenient tool because their dynamical evolution in the Schrödinger equation can be solved analytically. This is because any 16 17

H Aref, Vortices and polynomials, Fluid Dynamics Research 39, 5 (2007). D M Paganin, Coherent X-Ray Optics, Oxford University Press (2006).

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such wave function is a polynomial, which means that the action of the Taylor series expansion of the propagator (exponential of the Laplacian kinetic energy operator) will terminate at a finite order.

5.8 Coherence vortex FRS: coherence vortices18,19,20 Coherence vortices are phase singularities in generic complex-valued n-point correlation functions such as the density matrix. As such, coherence vortices are not restricted by the dimensionality of space the way the usual vortices are. For example, considering a 1D quantum system described by the field Ψ(x , t ), a quantised circulation of coherence

∮ ∇⊥ arg[g (1)(x, x′)] · dl⊥ = 2πw

(5.16)

defines a 1D coherence vortex, where the two-point equal-time correlation function is

g (1)(x , x′) =

〈Ψ †(x )Ψ(x′)〉 〈∣Ψ(x )∣2 〉〈∣Ψ(x′)∣2 〉

.

(5.17)

5.9 Fractional vortex FRS: fractional vortices21,22,23 In multicomponent systems, such as spinor BECs, vortices whose circulation is not an integer but a rational fraction of the circulation quantum may exist. For the purpose of demonstration, consider a half-quantum vortex state in a spin-1 system expressed by a three-component spinor wave function:

⎛ ψ+1(r)⎞ ⎛ iθ ⎞ ⎜ ⎟ 1 ⎜e ⎟ Ψ(r) = ⎜ ψ0(r) ⎟ = 0 . 2 ⎜⎝ 1 ⎟⎠ ⎜ ⎟ ⎝ ψ−1(r)⎠

(5.18)

18 G Gbur and T D Visser, Chapter 5 - The Structure of Partially Coherent Fields, Progress in Optics 55, 285 (2010). 19 T P Simula and D M Paganin, Coherence vortices in one spatial dimension, Physical Review A 84, 052104 (2011). 20 T P Simula and D M Paganin, Coherence simplices, New Journal of Physics 14, 113015 (2012). 21 H Mäkelä Y Zhang K -A Suominen, Topological defects in spinor condensates, Journal of Physics A: Mathematical and General 36, 8555 (2003). 22 Y Kawaguchi and M Ueda, Spinor Bose-Einstein condensates, Physics Reports 520, 253 (2012). 23 J A Sauls, Viewpoint: Half-Quantum Vortices in Superfluid Helium, Physics 9, 148 (2016).

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This may be viewed as a composite object with a plain single quantum vortex in component ψ+1, zero population in the ψ0 component, and a constant density in the ψ−1 component. The circulation of superflow is quantised in half-integer units of the circulation quantum 1 (5.19) vs · dℓ = κ 2 Γ

∮

and on traversing around the vortex, the spin vector d rotates π radians:

∮Γ ∇α · dℓ = π,

(5.20)

where α is the angle between the local spin vector and a reference unit vector. Straightforward extension to higher spin systems yields fractional vortices with other values of fractional circulation of mass and spin currents. For example, spin-2 BECs support fractional 1/3 and 2/3 vortices in addition to half-quantum vortices.

5.10 Baby skyrmion FRS: skyrmion24, baby skyrmion25, coreless vortices26,27 A baby skyrmion is a topological excitation in a (2+1)-dimensional simplification of the Skyrme model of nuclear physics. A spinor wave function

⎛ ⎞ cos(β 2)2 ⎜ ⎟ Ψ(r) = ⎜ e iθ 2 sin(β 2) cos(β 2)⎟ ⎜ ⎟ e i 2θ sin(β 2)2 ⎝ ⎠

(5.21)

parametrised in terms of the bending angle β (r) ∈ [0, π ] has the winding number combination (0, 1, 2). This results in a spin texture

〈F〉 = sin(β ) cos(θ )eˆ x + sin(β ) sin(θ )eˆ y + cos(β )eˆ z

(5.22)

and superfluid velocity field

T H R Skyrme, A non-linear field theory, Proceedings of the Royal Society London A 260, 127 (1961). B M A G Piette, B J Schroers, and W J Zakrzewski, Dynamics of baby Skyrmions, Nuclear Physics B 439, 205 (1995). 26 T -L Ho, Spinor Bose Condensates in Optical Traps, Physical Review Letters 81, 742 (1998). 27 T Mizushima, N Kobayashi and K Machida, Coreless and singular vortex lattices in rotating spinor Bose– Einstein condensates, Physical Review A 70, 043613 (2004). 24 25

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Quantised Vortices

vs =

ℏ 1 − cos(β ) eˆ θ. m r

(5.23)

Depending on the value of the bending angle β (R ) at the boundary R of the system, such a coreless vortex with β (0) = 0 is also known as a Mermin–Ho vortex if β (R ) = π /2 or Anderson–Toulouse vortex if β (R ) = π .

5.11 Monopole FRS: Dirac monopole28, abelian monopole29, non-abelian monopole30 In two-dimensional systems, both vortices and monopoles are point-like objects. In three-dimensional systems monopoles of the electric kind are abundant whereas monopoles of the magnetic kind have not been observed. Dirac showed that the existence of magnetic monopoles would not contradict the principles of quantum mechanics. Many flavours of artificial monopoles can be realised in many systems including spinor BECs. Notwithstanding, such synthetic monopoles do not correspond to an elementary particle referred to as a magnetic monopole.

5.12 Fluxon, chargeon, and dyon FRS: discrete gauge theories31, particles of the quantum double32,33 The algebraic structure of the Drinfeld or quantum double construction of quantum groups defines particle types of discrete gauge theories. The transformation matrices that represent fluxes (vortices) form a group, the stabiliser group H. A specific flux can be represented by a particular transformation or by any of its conjugates within this group. Fluxons are distinguishable flux types and are labelled by the sets of conjugate transformations—the conjugacy classes of H. In addition to fluxons (particles of magnetic-flux type), there are chargeons (particles of electric-charge type), labelled by representations of H. Dyons are charge–flux composites that carry both an electric charge and a magnetic flux and are labeled by a flux type (conjugacy class) and a representation of a subgroup of H—the centralizer of the flux. 28 P A M Dirac, Quantised singularities in the electromagnetic field, Proceedings of the Royal Society London Series A 133, 60 (1931). 29 M W Ray, E Ruokokoski, S Kandel, M Möttönen, D S Hall, Observation of Dirac monopoles in a synthetic magnetic field, Nature 505, 657 (2014). 30 S Sugawa, F Salces-Carcoba, A R Perry, Y Yue, I B Spielman, Second Chern number of a quantum-simulated non-Abelian Yang monopole, Science 360, 1429 (2018). 31 M de Wild Propitius and F A Bais, Discrete gauge theories, in G Semenoff and L Vinet Eds, Particles and Fields, CRM Series in Mathematical Physics, Springer (1998). 32 J Preskill, Lecture notes for physics 219: Quantum computation (2004). 33 E Witten, Dyons of charge eθ 2π , Physics Letters B 86, 283 (1979).

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5.13 Alice vortex and Cheshire charge FRS: Alice vortices34,35, Cheshire charges36,37 Two non-abelian vortices may carry a completely delocalised Cheshire charge which is not a property belonging to either one of the two vortices. However, such a delocalised charge has a long-range topological influence on other particles in the system. For example, upon traversing a Cheshire charge, a delocalized chargeon associated with a pair of vortices, around a pure fluxon it experiences an Aharonov– Bohm interaction with the fluxon. The Cheshire charge may be revealed (localised) upon fusion (merging) of the two vortices in the Cheshire charge state. Such vortices comprising a Cheshire charge are called Alice vortices/strings or an Alice ring if their ends are joined to form a closed loop.

34 J Preskill and L M Krauss, Local discrete symmetry and quantum-mechanical hair, Nuclear Physics B 341, 50 (1990). 35 J Ruostekoski and J R Anglin, Monopole Core Instability and Alice Rings in Spinor Bose–Einstein Condensates, Physical Review Letters 91, 190402 (2003). 36 M G Alford, K Benson, S Coleman, J March-Russell, and F Wilczek, Interactions and excitations of nonAbelian vortices, Physical Review Letters 64, 1632 (1990). 37 T Mawson, T Petersen, and T Simula, Braiding and fusion of non-Abelian vortex anyons, arXiv:1805.10009, (2018).

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Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 6 Structure of a plain vortex

6.1 Vortex uncertainty principle FRS: uncertainty in the vortex position1,2,3 A ‘plain’ vortex comprises two inherent components: (i) the shape of the condensate wavefunction, which has a circulating current around a phase singularity at a location where the condensate probability density is identically zero, and (ii) kelvons, quasiparticle bound states localized in the vicinity of the vortex core whose size is usually determined by the healing length ξ. In the core region, the condensate density ∣ϕ(r)∣2 ≈ r 2 forms a harmonic oscillator potential. The eigenstates of this potential are the kelvons with excitation energies Ek = ℏωk . The superflow circulates around the vortex phase singularity with superfluid velocity vs ∝ w /r . The Heisenberg uncertainty principle

Δpv Δxv ≈ ℏ 2

(6.1)

for the vortex may be expressed in terms of the inertial mass of the vortex m v as

(m vωkξ ) × (ξ ) ≈ ℏ 2,

(6.2)

which is a statement that the vortex has a momentum uncertainty determined by the speed uncertainty of the vortex phase singularity and a position uncertainty equal to the size of the vortex core. In other words, it is not possible to locate the position of a vortex particle more accurately than the region determined by the size of its core.

1

W F Vinen, Chapter I Vortex Lines in Liquid Helium II, Progress in Low Temperature Physics 3, 1 (1961). E Byckling, Vortex lines and the λ-transition, Annals of Physics 32, 367 (1965). 3 T Simula, Vortex mass in a superfluid, Physical Review A 97, 023609 (2018). 2

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6.2 Kelvon FRS: kelvon quasiparticle4,5

In a three-dimensional Bose–Einstein condensate (BEC) a single vortex has a branch of Kelvin modes characterised by their axial wave number. In two-dimensional (2D) BECs the axial bending modes of the vortex are suppressed and only one low-energy kelvon quasiparticle exists corresponding to the ground state kelvon with zero axial wave number. The physics of the ground state kelvon may be clearly illustrated by considering an axisymmetric single quantum vortex in a simple BEC at zero temperature. The condensate density at the vortex core is a quadratic function of position and forms a harmonic oscillator potential

V (ϕ(r < ξ )) =

1 mk ω k2r 2 , 2

(6.3)

for the kelvon quasiparticle. The kelvon is a vortex core localized bound state of such a potential and has an effective mass mk, a zero point energy Ek = ℏωk and an orbital angular momentum quantum number ℓ = −1 with respect to the condensate. The quasiparticle amplitudes of the kelvon are

uk(r ) = ∣uk(r )∣e i (ℓ+w )θ ,

and

vk (r ) = ∣vk (r )∣e i (ℓ−w )θ ,

(6.4)

where w is the winding number of the vortex and θ is the polar angle. At zero temperature, the kelvon has Bogoliubov–de Gennes (BdG) quasiparticle mode densities

∣uk∣2 ∝ r∣ℓ+1∣,

and

∣vk ∣2 ∝ r∣ℓ−1∣.

(6.5)

The kelvon is a source of small amplitude periodic cyclotron motion of the vortex core. In the literature the kelvon is sometimes referred to as the lowest core localized state, an anomalous mode, or the vortex precession mode. Within the Thomas– Fermi approximation the frequency of the kelvon for an axially symmetric vortex in a harmonic trap equals the metastability frequency of the vortex:

∣ωk∣ = Ωm =

⎛R ⎞ 3 ℏ ln ⎜ ⊥ ⎟. 2 2 MR ⊥ ⎝ ξ ⎠

(6.6)

In a uniform trap the kelvon frequency may be estimated by the result from classical fluid dynamics:

∣ωk∣ = Ωm =

4

ℏ . MR2

(6.7)

A L Fetter, Rotating trapped Bose–Einstein condensates, Reviews of Modern Physics 81, 647 (2009). T P Simula, T Mizushima and K Machida, Kelvin Waves of Quantized Vortex Lines in Trapped Bose–Einstein Condensates, Physical Review Letters 101, 020402 (2008). 5

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Quantised Vortices

Both of these results ignore the structure of the vortex core. In general, both the shape of the vortex core and kelvon quasiparticle mode, which is inherently related to the inertial mass of a vortex and its spin and dipole moment, should be determined self-consistently. The Caroli–de Gennes–Matricon (CdGM) modes in Fermi systems are the counterpart of the kelvons in Bose systems.

6.3 Circulation quantum FRS: quantised circulation6 A scalar vortex (a phase singularity in a complex valued field) is a source of a superfluid velocity field vs = wℏ∇θ /msf and results in quantisation of the circulation

Γ=

∮ vs · dℓ = wκ,

(6.8)

where w is the integer winding number of the vortex and the quantum of circulation

κ = h msf ,

(6.9)

with msf the mass of the particle responsible for the superfluid behaviour and h = 2π ℏ is Planck’s constant. In a neutral bosonic superfluid such as an atomic BEC or a superfluid helium-4, the mass msf = m is the mass of the atom responsible for the superfluidity such as a rubidium or helium atom. In neutral Fermi superfluids the mass msf = 2m is twice the mass of the fermionic particle forming (half of) the Cooper pair. In generic systems described by multicomponent order parameters the circulation is in general not quantised.

6.4 Vortex energy FRS: energy of a vortex7 The total energy of the vortex equals the energy cost to create it. This is often estimated to be equal to the incompressible kinetic energy added to the fluid. Since the velocity field (in the vortex centred coordinates) around an ideal vortex is vs = ℏ/mr , where r is the distance from the vortex core, the kinetic energy per unit length of a static vortex is

6 7

L Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6, 279 (1949). C J Pethick and H Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press (2008).

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Quantised Vortices

Ev =

1 2

∫ ρs ∣vs∣2 dxdy =

π ℏ2ρs ⎛ R ⎞ ln ⎜ ⎟ , ⎝ rc ⎠ m2

(6.10)

which assumes uniform superfluid density ρs = mn(r ). Formally, this energy becomes infinite in the limit R /rc → ∞ and reflects the topological protection of such a vortex as removing or inserting a single vortex in a uniform (infinite) system involves a change of the topology of the system—a deformation which costs an infinite amount of energy. However, in realistic situations vortices are always produced in vortex–antivortex pairs, which conserves the total circulation.

6.5 Thermodynamic stability FRS: thermodynamic critical frequency8 The stability and metastability of a vortex may be discussed in terms of its intrinsic structural integrity as well as the energetic considerations. Conservation laws due to symmetries present in the physical system may result in long-lived metastable states even if such states would otherwise be energetically disfavored. A vortex is said to be thermodynamically stable if the vortex state corresponds to an absolute minimum configuration of the Hamiltonian. In a simple superfluid this requires the system to be rotated at or above the thermodynamic critical frequency

Ωcth =

ΔE , ΔL

(6.11)

where ΔE = E1 − E0 is the energy difference between the state with and without the vortex, respectively, and ΔL = L1 − L 0 is the corresponding difference in orbital angular momentum. The argument is generic and straightforwardly extends to multiple circulation quanta in which case ΔE = Eℓ+1 − Eℓ and ΔL = L ℓ+1 − L ℓ are the cost of energy, and the change of orbital angular momentum of the system, respectively, when one quantum of vorticity is added to the existing system with ℓ circulation quanta. Considering a single quantum vortex in the centre of a harmonic trap that has ΔL = ℏ yields

Ωcth =

8

⎛R ⎞ 5 ℏ ln ⎜ ⊥ ⎟. 2 2 MR ⊥ ⎝ ξ ⎠

(6.12)

C J Pethick and H Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press (2008).

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Quantised Vortices

6.6 Spectral, energetic stability FRS: vortex nucleation frequency9 If a vortex state is thermodynamically stable, it also implies its excitation spectrum to be positive definite. However, a vortex in the absence of external rotation may have one or more real, negative energy eigenstates with respect to the chemical potential of the condensate. These ‘anomalous’ negative energy modes correspond to the vortex-core-localised kelvons, which in a dissipative system cause the vortex to spiral outward to the periphery of the system. However, if population transfer to these modes is forbidden, for example due to conservation of angular momentum, the vortex may exist in a long-lived metastable state despite being spectrally unstable. The critical frequency to reach spectral metastability, that is, to lift the negative energy kelvon mode to positive energy with respect to the chemical potential, is

Ωcm =

Em , ℓm

(6.13)

where Em is the excitation of the most negative energy excitation mode and ℓm is its orbital angular momentum with respect to the condensate. For an axisymmetric vortex in a simple BEC in a harmonic trap

Ωcm = ∣ωk∣.

(6.14)

The thermodynamic critical temperature

Ωcth > Ωcm

(6.15)

such that the spectral stability is a necessary condition for thermodynamic stability.

6.7 Dynamical Lyapunov stability FRS: dynamical stability10 The most severe requirement on the vortex stability is its dynamical stability, which may be characterized by Lyapunov exponents. The vortex state is dynamically unstable if its excitation spectrum contains one or more eigenstates with non-zero imaginary component, Im(Eq) ≠ 0. This implies that the population of such a destabilizing eigenstate may grow exponentially rapidly. Typically, the presence of dynamical instabilities occur due to a resonance between two quasiparticle states. 9 D L Feder, A A Svidzinsky, A L Fetter, and C W Clark, Anomalous Modes Drive Vortex Dynamics in Confined Bose–Einstein Condensates, Physical Review Letters 86, 564 (2001). 10 J J García-Ripoll and V M Pérez-García, Stability of vortices in inhomogeneous Bose condensates subject to rotation: A three-dimensional analysis, Physical Review A 60, 4864 (1999).

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Quantised Vortices

For example, a quantized vortex with p > 1 circulation quanta is dynamically unstable and will split up into p single quantum vortices via resonant coupling with a surface mode with a specific multipolarity. The dynamically unstable states are typically high-energy configurations, which may lower their free energy by changing their intrinsic structure.

6.8 Inertial vortex mass FRS: inertial vortex mass11,12,13,14,15,16,17,18,19,20 The problem of the mass of a vortex in a superfluid has a long history and over the decades several candidates have been proposed. Consider first a hollow or solid aluminium cylinder of radius rcyl moving through a classical incompressible inviscid fluid. The effective mass of the cylinder has two contributions. The first is the bare mass (per unit length) of the cylinder 2 mbare = πrcyl ρcyl = Ncylm cyl ,

(6.16)

which is simply the physical mass of the aluminium cylinder comprising Ncyl aluminium atoms of mass mcyl . The second contribution is the hydrodynamic induced mass 2 m induced = πrcyl ρfluid ,

(6.17)

which is equal to the physical mass of the fluid displaced by the cylinder. This second mass contribution is due to the motion of the fluid that needs to flow back to fill in the hole left behind where the cylinder used to be. For a hollow core vortex the induced mass is equal to a bare mass of a virtual fluid filled core:

11

V N Popov, Quantum vortices and phase transitions in Bose systems, Soviet Physics JETP 37, 341 (1973). N B Kopnin, Frequency singularities of the dissipation in the mixed state of pure type-II superconductors at low temperatures, Journal of Experimental and Theoretical Physics Letters 27, 390 (1978). 13 G Baym and E Chandler, The hydrodynamics of rotating superfluids. I. Zero-temperature, nondissipative theory, Journal of Low Temperature Physics 50, 57 (1983). 14 J -M Duan and A J Leggett, Inertial mass of a moving singularity in a Fermi superfluid, Physical Review Letters 68, 1216 (1992). 15 J H Han, J S Kim, M J Kim and P Ao, Effective vortex mass from microscopic theory, Physical Review B 71, 125108 (2005). 16 D J Thouless and J R Anglin, Vortex Mass in a Superfluid at Low Frequencies, Physical Review Letters 99, 105301 (2007). 17 G E Volovik, The Universe in a Helium Droplet, Oxford University Press (2009). 18 A Klein, I L Aleiner, O Agam, The internal structure of a vortex in a two-dimensional superfluid with long healing length and its implications, Annals of Physics 346, 195 (2014). 19 L A Toikka and J Brand, Asymptotically solvable model for a solitonic vortex in a compressible superfluid, New Journal of Physics 19, 023029 (2017). 20 T Simula, Vortex mass in a superfluid, Physical Review A 97, 023609 (2018). 12

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Quantised Vortices

hollow 2 virtual m induced = πrcyl ρfluid = m bare .

(6.18)

A rotating cylinder placed in a fluid provides a naïve picture of a vortex in a superfluid. However, the general picture remains the same for real vortices in that the effective mass of the vortex has in general two contributions—one due to the matter filling the vortex core and another one due to the motion of the embedding fluid displaced by the vortex.

6.9 Gravitational vortex mass FRS: gravitational vortex mass21 The inertial vortex mass may be defined by the Newton’s law, FI = m Iv a , as the constant of proportionality between the acceleration a of the vortex and the force FI that causes it. Similarly, the gravitational vortex mass may be defined by the Newton’s law, FG = m Gv g , as the constant of proportionality between the acceleration g of the vortex and the gravitational force FG that causes it. Although in general these two masses could be unrelated, in a simple 2D superfluid universe the emergent gravitational and inertial vortex masses have the same origin.

6.10 Kelvon-based vortex mass FRS: kelvon based vortex mass22 The total (kelvon-based) inertial rest mass of a vortex in a Bose superfluid has been proposed to be equal to

m0 =

2π ℏn 0 , ωk

(6.19)

where n0 is the embedding condensate density (the condensate particle density in the absence of the vortex) and ωk is the excitation frequency of the kelvon quasiparticle. It includes all mass contributions such as core filling matter and quasiparticles as well as hydrodynamic induced mass effects. Obtaining the value for ωk , and thereby for m0, requires a self-consistent solution to the BdG equations. A classical limit of large quantum numbers (w → ∞) may be obtained by replacing the quantum of circulation κ = h /m by the classical circulation Γ = wκ and the kelvon frequency ωk by the frequency of the classical Kelvin wave frequency ω K . This yields an inertial mass of a classical vortex:

21 22

T Simula, Gravitational vortex mass in a superfluid, (2019). T Simula, Vortex mass in a superfluid, Physical Review A 97, 023609 (2018).

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Quantised Vortices

mv =

Γρ , ωK

(6.20)

where ρ is the mass density of the fluid.

6.11 Hydrodynamic induced vortex mass component FRS: added mass23 The hydrodynamic induced (added) mass of a plain vortex

m induced =

∫ ( ρs − ρv )da,

(6.21)

where ρs and ρv are, respectively, the areal fluid density in the absence and presence of the vortex, is equal to the virtual bare mass of the displaced atoms, that is, the mass of the atoms displaced by the vortex core. For a hard core cylindrical vortex core shape

m induced = πrc2ρs ,

(6.22)

whereas for a more realistic soft core shape

m induced = ρs

∫r

c

R

⎛ ⎜1 − ⎜ ⎝

⎞ 2 ⎛ ⎞2 ⎟ 2πrdr = πrc2ρ ln ⎜ 1 R + 2 ⎟ , s ⎟ 3⎠ ⎝ 3 rc2 r 2 + 2rc2 ⎠ r2

(6.23)

which features a logarithmic divergence.

6.12 Relativistic vortex mass component FRS: relativistic vortex mass24 The ‘relativistic’ mass of a vortex

mfield =

κ 2ρs ⎛ R ⎞ Ev ln ⎜ ⎟ , = 2 ⎝ rc ⎠ 2π cs

(6.24)

equals the (incompressible kinetic) energy cost to create a vortex divided by the square of the speed of sound cs. This vortex mass candidate is logarithmically divergent even for a finite size system in the limit rc /R → 0 of a vanishingly small vortex core. 23

G G Stokes, On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, Transactions of the Cambridge Philosophical Society 9, 8 (1851). 24 V N Popov, Quantum vortices and phase transitions in Bose systems, Soviet Physics JETP 37, 341 (1973).

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Quantised Vortices

6.13 Baym–Chandler vortex mass FRS: Baym-Chandler mass25 The Baym–Chandler mass of a vortex in a Bose superfluid is

m BC =

2π ℏn 0 cylinder = πrc2ρs = m induced , ωI

(6.25)

where ωI ≫ ωk is the frequency of the inertial wave of a rotating fluid. For helium II the Baym–Chandler mass equals the hydrodynamic induced mass of a vortex cylinder, which in that case is also equal to the virtual bare mass of such a vortex.

6.14 Kopnin vortex mass FRS: Kopnin mass26,27 The Kopnin mass of a vortex in a Fermi superfluid is

m Kopnin =

π ℏn 0 , ωf

(6.26)

where ωf is the frequency of the level spacing of the CdGM vortex core modes in a fermion superfluid. For singular vortices

ωf ≈

1 Δ2 , 2 EF

(6.27)

where Δ is the gap function and EF is the Fermi energy, the Kopnin mass is approximately equal to the induced vortex mass.

25

G Baym and E Chandler, The hydrodynamics of rotating superfluids. I. Zero-temperature, nondissipative theory, Journal of Low Temperature Physics 50, 57 (1983). 26 N B Kopnin, Frequency singularities of the dissipation in the mixed state of pure type-II superconductors at low temperatures, Journal of Experimental and Theoretical Physics Letters 27, 390 (1978). 27 G E Volovik, The Universe in a Helium Droplet, Oxford University Press (2009).

6-9

IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 7 Vortex dynamics

7.1 Adiabatic vortex dynamics FRS: adiabaticity criterion for vortices1,2,3 The adiabatic theorem of quantum mechanics states that a physical system remains in its instantaneous eigenstate if a perturbation acts on the system sufficiently slowly such that excitations between different eigenstates of the non-degenerate spectrum of the system are improbable. When applied to motion of a vortex in a superfluid, this means that the vortex must move sufficiently slowly so that it does not significantly radiate (Bogoliubov) excitations that could, for example, lead to production of vortex–antivortex pairs in its wake, thus changing the many-body eigenstate of the system. However, the adiabaticity criterion for vortex motion is in general more stringent than the requirement of the vortex speed to remain below the vortex shedding critical velocity. The adiabaticity criterion of moving vortices will be an important consideration in topological quantum computers based on braiding nonabelian vortex anyons.

1

M Born and V Fock, Beweis des Adiabatensatzes, Zeitschrift fur Physik 51, 165 (1928). E Šimánek, Adiabatic criterion for fermion bound states in the core of a moving vortex, Physical Review B 46, 14054 (1992). 3 S M M Virtanen, T P Simula and M M Salomaa, Adiabaticity Criterion for Moving Vortices in Dilute Bose– Einstein Condensates, Physical Review Letters 87, 230403 (2001). 2

doi:10.1088/2053-2571/aafb9dch7

7-1

ª Morgan & Claypool Publishers 2019

Quantised Vortices

7.2 Vortex force and velocity FRS: force on a vortex4,5 The velocity of a plain vortex in a simple superfluid equals

vv = vs + vn =

∇es ℏ ℏ eˆ z × ∇S0(r , t ) − m 2m es r=rv

,

(7.1)

r=rv

where S0(r, t ) = S (r, t ) − S1(r, t ) is the embedding phase of the condensate after the phase of the vortex S1(r, t ) = arctan( y − yv , x − xv ) has been removed from the total condensate phase S (r, t ). The embedding function es(r ) is defined in terms of the condensate density n(r ) = (r − rv )2es(r ). The first term in equation (7.1) is the superfluid velocity vs at the location of the vortex produced by the other (image or real) vortices and the second term vn is due to the buoyancy effect caused by the curvature of the background condensate density at the location of the vortex phase singularity. The total force acting on a vortex is

Fv = −κmns eˆ z × vv ,

(7.2)

where ns is the condensate density at the location of the vortex in the absence of the vortex, and it has the two aforementioned contributions, one due to the background superfluid phase gradient and one due to the background density gradient. The Magnus force is

FM = −κmns eˆ z × vn.

(7.3)

If the condensate density is uniform the Magnus force vanishes and the vortex moves at the local superfluid velocity, vv = vs as if it was frozen in the fluid. The vortex force and the geometric phase γC accumulated as the vortex is transported along a closed loop C are related by

γC =

∮C

Fv ℏvv

dA = 2πNC ,

(7.4)

where NC is the number of condensate atoms enclosed by the vortex trajectory.

4

A J Groszek, D M Paganin, K Helmerson and T P Simula, Motion of vortices in inhomogeneous Bose– Einstein condensates Physical Review A 97, 023617 (2018). 5 T Simula, Vortex mass in a superfluid, Physical Review A 97, 023609 (2018).

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Quantised Vortices

7.3 Magnus effect and mutual induction FRS: Magnus effect6 A simple vortex is a source of a circulating velocity field around its core of the form

vs =

ℏ κ 1 ∇S ( r ) = eˆ ϕ. m 2π ∣r − rv∣

(7.5)

A Magnus force acts on the vortex perpendicular to the relative velocity

FM = −κmns eˆ z × (vv − vs ) = π

ℏ2 ns eˆ z × (eˆ z × ∇es ) . 2m es r=rv

(7.6)

In a uniform system FM = 0 and the vortex moves with the velocity, vv = vs , of the fluid it is embedded in. However, if the superfluid is non-uniform with ∇es ≠ 0, then the Magnus force is non-zero and the vortex will move with a relative velocity with respect to the fluid. Two initially immobile vortices vv = 0 separated by a distance d in a uniform superfluid induce motion on each other. The force acting on vortex 1 due to the superflow induced by vortex 2 is

F1 = −κ1mns eˆ z × vs 2

r=r1

= −mns

κ1κ2 1 eˆ 12. 2π d

(7.7)

The force acting on vortex 2 due to the superflow induced by vortex 1 is

F2 = −κ2mns eˆ z × vs1

r=r 2

= −mns

κ1κ2 1 eˆ 21. 2π d

(7.8)

Therefore, for κ1 = κ2 the vortices interact repulsively, whereas for κ1 = −κ2 the vortex–vortex force is attractive. If the vortices are free to move, the same sign vortices will rotate about their mutual centre of mass while a vortex–antivortex pair will translate with a constant velocity.

7.4 Vortex pair creation and annihilation FRS: vortex dipoles7 Plain vortices are topologically protected in the sense that an isolated vortex cannot be locally created or removed from the superfluid since it would violate the

6 H G Magnus, Ueber die Abweichung der Geschosse, und: Ueber eine auffallende Erscheinung bei rotirenden Körpern, Annalen der Physik 164, 1 (1853). 7 T W Neely, E C Samson, A S Bradley, M J Davis, and B P Anderson, Observation of Vortex Dipoles in an Oblate Bose–Einstein Condensate, Physical Review Letters 104, 160401 (2010).

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Quantised Vortices

conservation law of the circulation. Vortices may therefore only be created or annihilated in pairs to conserve the total topological flux (circulation) of the system. When a vortex and an antivortex annihilate, their internal energy is released via phonon emission. At zero temperature a single vortex–antivortex pair cannot be annihilated in the absence of dissipative effects and instead the pair will simply travel at a constant average speed and separation. However, if at least one more vortex (three in total with at least one vortex of each sign of circulation) is present, the vortex motion may become chaotic and the third vortex may also catalyse the annihilation process by absorbing the energy. Prior to annihilation, a vortex– antivortex pair may form a Jones–Roberts soliton, also called a vortexonium bound state, in which the distance between the two vortices is of the order of their core size. In this case the phase singularities associated with the two vortices disappear and the localised perturbation travels approximately at the speed of sound through the system.

7.5 Onsager point vortex model FRS: point vortex model8,9 A vortex is a source of a superfluid velocity field vsi = wκ /(r − ri). The total superfluid velocity field of a collection of N vortices is therefore Vtot = ∑i vsi . Hence the total kinetic energy associated with the superflow of an assembly of vortices within the point vortex approximation, which excludes the vortex cores, is

Esf =

1 2

∫ ρs ∣Vtot∣2 dA.

(7.9)

This may be expressed as a pseudo-Hamiltonian: N

H 2D =

∑ i=1

pi 2 2m v

N −1 N

−

ρs ∑ 2π i = 1

∑ sisj ln(∣ζi − ζj∣),

(7.10)

j=i+1

i where pi = m vvrel is the momentum of the vortex of inertial mass m v and ζi is a twodimensional (2D) vector of phase-space coordinates. The vortex kinetic energy (the first term) is usually ignored on the grounds that the mass of a vortex would be zero such that pi = 0. However, pi ≈ 0 not because of the inertial mass of a vortex, which in general is large, but instead because the velocity of the vortex vv with respect to the velocity vs of the superfluid, vrel = (vv − vs ), is vanishingly small in uniform incompressible fluids in which the vortices tend to move at the local superfluid velocity. Nevertheless, under certain circumstances, such as in the case of quantum turbulence and at high temperatures, the effects of compressibility should not be 8

C C Lin, On the motion of vortices in two dimensions. I. Existence of the Kirchhoff–Routh function, PNAS 27, 570 (1941). 9 L Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6, 279 (1949).

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Quantised Vortices

ignored due to the presence of large density fluctuations. Such conditions result in significant relative velocities such that neglecting the kinetic energy associated with the inertia of the vortices is not well founded. The second term in the pseudoHamiltonian is the logarithmic vortex–vortex interaction potential. For vrel = 0 the dynamics of a point vortex assembly is determined by the equations

κsi

dxi ∂H 2D = dt ∂yi

(7.11)

κsi

dyi ∂H 2D . =− dt ∂xi

(7.12)

Dissipation may be included in the vortex system by adding a phenomenological damping term that breaks the energy conservation. Another form of vortex dissipation may also be included by explicitly removing vortex–antivortex pairs whenever they approach within distance dann of each other and this breaks the particle number conservation.

7.6 Vortex–particle duality FRS: vortex-particle duality10 The equations of motion for a vortex in a 2D incompressible fluid may be expressed in terms of an effective, inverted, one-dimensional (1D) harmonic oscillator Hamiltonian:

−H

1D

=

py2 2m v

+

1 m vω 2qx2 + Uint. 2

(7.13)

The equations of motion are

∂qx ∂H1D = ∂py ∂t ∂py ∂t

=−

∂H1D ∂qx

(7.14)

(7.15)

where qx and py are associated with the 2D Cartesian coordinates of the vortex in the 2D fluid. These equations are mathematically equivalent to Hamilton’s equations of a massive particle in 1D space. Equivalently, the dynamics of the vortices in real 2D

10 R N Valani, A J Groszek, and T P Simula, Einstein–Bose condensation of Onsager vortices, New Journal of Physics 20, 053038 (2018).

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Quantised Vortices

space may be interpreted as the phase-space dynamics of a particle in one spatial dimension. This constitutes the vortex–particle duality, wherein the dynamics of a vortex singularity in 2D corresponds to a phase-space dynamics of a vortex particle in 1D.

7.7 Point vortex model with cylindrical boundary FRS: point vortex model in a disk11 For an impenetrable cylindrical boundary with unit radius the pseudo-Hamiltonian is N N

H=−

N N

ρs κ 2 ρ κ2 si sj ln(rij2 ) + s ∑∑si sj ⎡⎣ln(d ij2 ) + ln(rj2 )⎤⎦ , ∑∑ 8π i = 1 j ≠ i 8π i = 1 j = 1

(7.16)

where

rij =

(xi − xj )2 + (yi − yj )2

and

(xi − xˆj )2 + (yi − yˆj )2

dij =

(7.17)

are, respectively, the distance between two vortices and the distance between a vortex and an image vortex located at a position (xˆi = xi /ri2, yˆi = yi /ri2 ). All lengths are dimensionless. The first term in the Hamiltonian is the pairwise interaction of the system vortices. The second term is the interaction between the system vortices and the image vortices, including the selfies. The equations of motion for this system are

yi − yˆj ⎤ ⎥ d ij2 ⎥⎦

(7.18)

⎤ xi − xˆj ⎥ , d ij2 ⎥⎦

(7.19)

⎡N y −y dxi κ i j = − ⎢∑sj 2 − dt 2π ⎢⎣ j ≠ i rij

∑sj

⎡N xi − xj dyi κ = + ⎢∑sj − dt 2π ⎢⎣ j ≠ i rij2

∑sj

N

j=1

N

j=1

where the first terms are the resultant velocity field due to the other system vortices and the second terms are the resultant velocity field due to the image vortices. In addition to the energy H and the total circulation ∑i si , the cylindrical symmetry allows another, vortex moment of inertia, constant of motion: N

Lv =

∑si (xi2 + yi2 ).

(7.20)

i=1

Note that this quantity does not equal the orbital angular momentum of the fluid.

11 Y B Pointin and T S Lundgren, Statistical mechanics of two-dimensional vortices in a bounded container, The Physics of Fluids 19, 1459 (1976).

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Quantised Vortices

7.8 Point vortex models with square boundaries FRS: periodic boundary conditions12,13, square wall boundary conditions14 For a square domain of unit length with periodic boundary conditions in both Cartesian directions the point-vortex equations of motion may be expressed as N ∞ sin(yij ) dxi = −κ∑sj ∑ dt cosh(xij − 2πm) − cos(yij ) j ≠ i m =−∞

(7.21)

N ∞ sin(xij ) dyi , = +κ∑sj ∑ dt cosh( y 2 m ) cos( x ) − π − ij ij j ≠ i m =−∞

(7.22)

where xij = 2π (xi − xj ). The point-vortex energy of the configuration is N N

H=−

ρs κ 2 ∑∑h(xij , yij ) 2 i=1 j≠i

(7.23)

where ∞

h(x , y ) =

cosh(x − 2πm) − cos(y ) x2 . − cosh(2πm) 2π m =−∞

∑

(7.24)

Typically, it is sufficient to include only few terms in the above infinite sums over the image vortices to obtain reasonable numerical accuracy. Hard wall boundary conditions may be implemented simply, at the cost of computation, by dividing the square domain into four square cells and for each vortex located in the ‘first’ cell at (xi , yi ) placing an image vortex of the same sign at (1 − xi , 1 − yi ) and two image vortices of the opposite sign of circulation, one at (1 − xi , yi ) and another at (xi , 1 − yi ). As a result a hard wall box boundary condition for N vortices is realised in the first cell of the full domain at the cost of simulating an equivalent of 4N judiciously positioned vortices with periodic boundary conditions.

12 J B Weiss and J C McWilliams, Nonergodicity of point vortices, Physics of Fluids A: Fluid Dynamics 3, 835 (1991). 13 M T Reeves, T P Billam, X Yu, and A S Bradley, Enstrophy Cascade in Decaying Two-Dimensional Quantum Turbulence, Physical Review Letters 119, 184502 (2017). 14 L J Campbell and K O’Neil, Statistics of two-dimensional point vortices and high-energy vortex states, Journal of Statistical Physics 65, 495 (1991).

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Quantised Vortices

7.9 Point vortex models in general domains FRS: generalised boundary conditions15,16,17,18, inhomogeneous condensates19, spinors20, cylindrical surface21, spherical surface22 The point-vortex equations of motion in a generic domain D are

κsi

dxi ∂H ; = dt ∂yi

κsi

dyi ∂H , =− dt ∂xi

(7.25)

where the Kirchhoff–Routh function N

N N

H = κ∑si ψ0(xi , yi ) + κ 2∑∑si sj G (xi , yi ; xj , yj ) i=1

i=1 j≠i

(7.26)

N

1 − κ 2∑si2g(xi , yi ; xj , yj ), 2 i=1 is expressed in terms of the stream function ψ0 and the hydrodynamic Green’s function G. The harmonic function g is defined by

g(x , y ; x0, y0) = −G (x , y ; x0, y0) −

1 ⎡ ln⎣(x − x0)2 + (y − y0)2 ⎤⎦. 4π

(7.27)

Using a conformal map z(ζ ) from the, generally multiply connected, interior of a unitζ disc Dζ to the interior of a unit-z disc Dz, the Kirchhoff–Routh path function is N

(

)

(

)

H (z ) z1, z1*, … , zN , zN* = H (ζ ) ζ1, ζ1*, … , ζN , ζN* +

κ2 ∑si2 ln∣zζ (ζi )∣, 4π i = 1

(7.28)

15 C C Lin, On the motion of vortices in two dimensions. I. Existence of the Kirchhoff–Routh function, PNAS 27, 570 (1941). 16 D Crowdy and J Marshall, Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains, Proceedings of The Royal Society A 461, 2477 (2005). 17 T L Ashbee, J G Esler, and N R McDonald, Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions, Journal of Computational Physics 246, 289 (2013). 18 D G Dritschel and S Boatto, The motion of point vortices on closed surfaces, Proceedings Mathematical, Physical, and Engineering Sciences/The Royal Society 471, 20140890 (2015). 19 A J Groszek, D M Paganin, K Helmerson and T P Simula, Motion of vortices in inhomogeneous Bose– Einstein condensates, Physical Review A 97, 023617 (2018). 20 K Kasamatsu, M Eto, and M Nitta, Short-range intervortex interaction and interacting dynamics of halfquantized vortices in two-component Bose–Einstein condensates, Physical Review A 93, 013615 (2016). 21 N -E Guenther, P Massignan, and A L Fetter, Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli, Physical Review A 96, 063608 (2017). 22 R Kidambi, and P K Newton, Point vortex motion on a sphere with solid boundaries, Physics of Fluids 12, 581 (2000).

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Quantised Vortices

where zi and ζi are the positions of the vortices in the z and ζ domains, respectively, where a complex coordinate notation z = x + iy and z * = x − iy is used. Such conformal mappings may be used, for example, to map the unit disc onto an elliptical domain or more generically to multiply connected domains. Particularly interesting cases for point-vortex dynamics occurs on surfaces of cylinders and spheres and for vortices with internal spinor structure. Extending the point vortex models to situations where the uniformity of the fluid density is not a reasonable assumption due to the compressibility of the fluid is, in general, difficult.

7.10 Vortex classification algorithm FRS: clusters, dipoles and free vortices23,24 An arbitrary configuration of N+ vortices and N− antivortices may be uniquely divided into three categories: free vortices, vortex–antivortex dipoles and vortex clusters of either sign. Such a classification is achieved by an algorithm based on four steps: (i) measure all pairwise distances between vortices, (ii) a vortex and an antivortex form a dipole if they are mutual nearest neighbours, (iii) the vortices with same sign nearest neighbours belong to a same cluster, and (iv) all remaining vortices are free vortices.

7.11 Vortex temperature FRS: vortex thermometry25 In a simple scalar superfluid the Bogoliubov quasiparticles may be divided into two topologically distinct classes: (i) topologically trivial phonons associated with the compressible condensate density modulations, and (ii) topologically non-trivial, incompressible vortex waves associated with the motion of the quantised vortices. These two components may be associated with distinct temperatures, one for the phonons and one for the vortices, and in general non-equilibrium situations these two temperatures need not be equal. The phonon temperature typically corresponds to the usual thermodynamic temperature of the fluid. The vortex temperature is a statistical mechanical temperature defined entirely in terms of the vortex degrees of freedom. The fractional populations of vortex clusters Nc /Ntot , dipoles Nd /Ntot and free vortices Nf /Ntot may be used for determination of the temperature of the vortex gas. To determine the vortex temperature a vorticity field of a fluid is first 23 M T Reeves, T P Billam, B P Anderson, and A S Bradley, Inverse Energy Cascade in Forced TwoDimensional Quantum Turbulence, Physical Review Letters 110, 104501 (2013). 24 R N Valani, A J Groszek, and T P Simula, Einstein-Bose condensation of Onsager vortices, New Journal of Physics 20, 053038 (2018). 25 A J Groszek, M J Davis, D M Paganin, K Helmerson, and T P Simula, Vortex Thermometry for Turbulent Two-Dimensional Fluids, Physical Review Letters 120, 034504 (2018).

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approximated with a total number of Ntot = N+ + N− = Nc + Nd + Nf vortices and antivortices. In a BEC these are the real physical vortices whereas for a classical fluid whose circulation is not quantized these are effective vortices. The vortex temperature may be measured by the fraction of clustered vortices via relation

⎛N ⎞ Tv = f ⎜ c ⎟ , ⎝ Ntot ⎠

(7.29)

where f (x ) is a monotonic function of its argument that smoothly interpolates between zero and one, x ∈ [0, 1], and whose precise form may be calibrated by Monte-Carlo sampling of all possible vortex configurations for the system. Generically, depending on its vortex temperature, the neutral vortex gas may exist in three distinct phases: (i) a high-entropy, infinite-temperature, normal phase in which the number of clustered vortices, dipole vortices and free vortices are all comparable; (ii) a low-vortex energy, low-entropy phase where most of the vortices are bound in dipoles, separated by a Berezinskii–Kosterlitz–Thouless (Hauge– Hemmer) transition from the high-entropy normal phase; and (iii) a high-vortex energy, low-entropy phase where most of the vortices are found in clusters. The high-entropy normal state is separated from the vortex-cluster-dominated Onsager vortex state by the Onsager vortex condensation transition.

7.12 Winding number scaling laws FRS: winding number scaling laws26 The state of a vortex gas may also be characterized by the winding number fluctuations. If the spatial positions of vortices and antivortices in a neutral system are drawn from uniform random distribution, then the mean circulation 〈w(R◦)〉 = ∣N+ − N−∣ inside a circle of any radius R◦ is zero. However, the fluctuations δw 2 = 〈w 2(R◦) − 〈w(R◦)〉2 〉 are proportional to the area of the circle, δw 2 ∝ R◦2 , for R◦ > d v , where d v is the mean intervortex spacing. This is the case for the maximum entropy vortex configurations with approximately equal numbers of clusters, dipoles and free vortices. However, if the condensate phases, rather than the vortex positions, are randomised the vortices are found to be bound into vortex– antivortex pairs and the anti-correlations between vortex positions result in the winding number fluctuations to scale in proportion to the circumference of the circle δw 2 ∝ 2πR◦. This type of scaling is characteristic of the dipole-dominated, lowenergy, low-entropy states in the vicinity of the BKT transition and following a Kibble–Zurek quench. In the high-energy clustered states in the vicinity of the Onsager vortex transition, the strong positive correlations in vortex positions yield winding number scaling that fluctuates as a function of the system size.

26 W H Zurek, Topological relics of symmetry breaking: winding numbers and scaling tilts from random vortex– antivortex pairs, Journal of Physics Condensed Matter 25, 404209 (2013).

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IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 8 Vortex production in Bose–Einstein condensates

Quite generically, quantum vortices may be produced in ordered systems whenever sufficient amount of surplus energy is accessible for the system. Such conditions may be achieved either by supplying the required energy using external potentials or by changing (quenching) the system parameters such that the state of the system no longer corresponds to the energetically preferred (ground-)state of the Hamiltonian. The latter may result in a thermodynamic or quantum phase transition that involves spontaneous generation of topological defects. In both cases, excess energy becomes available for the system’s disposal and this extra energy may be spent by nucleating topological defects such as vortices. Due to these considerations, production of vortices in superfluids tends to be rather straightforward. In fact, in practical experiments on bulk superfluid helium it is nearly impossible to realise vortex-free states as there are always remnant vortices present in the system due to the small energy cost of a single vortex with respect to the bulk energy of the fluid. A list of vortex-generating experiments in Bose–Einstein condensates (BECs) can be found at http://www.becvortex.com/.

8.1 Coherent coupling of internal states FRS: phase engineering technique1 Coherent internal state engineering can be used for producing a vortex in two component BECs. For example, using 87Rb atoms and starting with all condensate atoms in the state ∣1〉 = (F = 1, mF = −1), an off-resonant microwave field can be used to provide coupling to another, ∣2〉 = (F = 2, mF = +1), internal state. A spatio-temporal modulation of an ac Stark shift provided by a suitably tuned 1 M R Matthews, B P Anderson, P C Haljan, D S Hall, C E Wieman, and E A Cornell, Vortices in a Bose– Einstein Condensate, Physical Review Letters 83, 2498 (1999).

doi:10.1088/2053-2571/aafb9dch8

8-1

ª Morgan & Claypool Publishers 2019

Quantised Vortices

focused laser beam then facilitates coherent transfer of atoms with orbital angular momentum ℓ = 0 from the state ∣1〉 to state ∣2〉 with orbital angular momentum ℓ = 1 ℏ. By controlling the duration of the applied electromagnetic fields, the number of atoms transferred to the vortex state can be varied.

8.2 Laguerre–Gauss laser modes FRS: Laguerre-Gauss technique2, photon angular momentum3 The amplitude of the electric field of Laguerre–Gauss (LG) laser modes (doughnut modes) have the form ⎛

LG mℓ(ρ ,

ϕ) =

⎛ 2ρ 2 ⎞ ρ ℓ −⎜⎜ ⎝ w0 e ⎟ ⎝ w 2(ζ ) ⎠ w(ζ ) ℓ+1

L mℓ⎜

ρ 1 +iζ

⎞2 ⎟⎟ ⎠ e ikze −iℓϕe −i ΦG ,

(8.1)

where L mℓ are the associated Laguerre polynomials, ϕ is the azimuthal coordinate, ρ = x 2 + y 2 is the radial coordinate and ζ = z /zR is the coordinate along the photons’ propagation axis scaled by the Rayleigh range, zR = 12 kw02 , where k = 2π /λ is the wave vector with λ the wavelength of the light. The beam waist w(ζ ) = w0 1 + ζ 2 , and the Gouy phase shift ΦG = (ℓ + 2m + 1) arctan(ζ ). The integers ℓ and m define the number or nodes in the azimuthal and radial directions, respectively. Hence, ℓ is also the winding number of such an optical vortex. For ℓ = 0 equation (8.1) reduces to a Gaussian (G) laser beam and for ℓ = 1 it yields a single quantum optical vortex. Generally a laser photon in the LG 0ℓ a mode carry ℓ units of orbital angular momentum, ℓ ℏ, which can be coherently transferred to a single atom in a BEC, for example, via a two-photon coherent, momentum conserving process in which the condensate absorbs an LG photon with linear momentum ℏk and an orbital angular momentum ℓ ℏ and subsequently relaxes via stimulated emission by emitting a G photon with momentum −ℏk and zero orbital angular momentum to a Gaussian laser beam. This leaves the condensate atom in a vortex state. Controlling the duration of the applied laser pulses enables one to vary the number of atoms transfered to the vortex state. Such a coherent photon transfer method can be straightforwardly extended in a multitude of ways by coupling the internal (spin) states of the atoms with photons of different spin (circular polarisation) and orbital angular momenta. Such techniques can be used for engineering traveling and stationary vortex states in condensates and for storage (stopping) of light.

2

M F Andersen, C Ryu, P Cladé V Natarajan, A Vaziri, K Helmerson, and W D Phillips, Quantized Rotation of Atoms from Photons with Orbital Angular Momentum, Physical Review Letters 97, 170406 (2006). 3 L Allen, S Barnett, M Padgett, Optical Angular Momentum, CRC Press (2003).

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Quantised Vortices

8.3 Topological angular momentum conversion FRS: topological phase imprinting4,5 In a spinor BEC, it is possible to use topological methods to generate vortices into the condensate wave function. Consider a spin-1 BEC in a magnetic quadrupole trap with bias magnetic field polarising the condensate such that all of the atoms are in the internal mf = 1 spin state. Adiabatically moving the zero of the magnetic field through the centre of the condensate, such that the local spin vector of the condensate follows the orientation of the local magnetic field, transfers all the atoms to the internal mf = −1 spin state. This results in a change Δmf = 2 of the spinangular momentum of each atom, and conservation of total angular momentum in the process requires the atoms at the end of the process to carry orbital angular momentum of 2ℏ, leaving the condensate in a double quantum vortex state. This process can be described in terms of a geometric Berry phase acquired by the condensate order parameter when an external parameter (magnetic bias field) of the Hamiltonian is varied. It also demonstrates, perhaps surprisingly, a topological equivalence between the non-vortex (zero orbital angular momentum) state and the double quantum vortex state in this system.

8.4 Rotating bucket FRS: small vortex lattices6,7, large vortex lattices8,9 Consider an ideal rotating bucket experiment where a weakly dissipative superfluid is confined by a bucket that is rotating at an angular frequency Ω. The Hamiltonian of the system in the rotating frame is

HΩ = H0 − Ω · L,

(8.2)

where H0 is the Hamiltonian in the laboratory frame of reference and L is the orbital angular momentum operator. In such an ideal rotating bucket experiment vortices are nucleated from the outside, at the boundary, of the condensate when the angular 4 A E Leanhardt, Y Shin, D Kielpinski, D E Pritchard, and W Ketterle, Coreless Vortex Formation in a Spinor Bose–Einstein Condensate, Physical Review Letters 90, 140403 (2003). 5 T Isoshima, M Nakahara, T Ohmi, and K Machida, Creation of a persistent current and vortex in a Bose– Einstein condensate of alkali-metal atoms, Physical Review A 61, 063610 (2003). 6 K W Madison, F Chevy, W Wohlleben, and J Dalibard, Vortex Formation in a Stirred Bose–Einstein Condensate, Physical Review Letters 84, 806 (2000). 7 E Hodby, G Hechenblaikner, S A Hopkins, O M Maragò and C J Foot, Vortex Nucleation in Bose–Einstein Condensates in an Oblate, Purely Magnetic Potential, Physical Review A 88, 010405 (2001). 8 J R Abo-Shaeer, C Raman, J M Vogels, W Ketterle, Observation of Vortex Lattices in Bose–Einstein Condensates, Science 292, 476 (2001). 9 P C Haljan, I Coddington, P Engels, and E A Cornell, Driving Bose–Einstein-Condensate Vorticity with a Rotating Normal Cloud, Physical Review Letters 87, 210403 (2001).

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speed of the walls of the bucket (confining external potential) exceed the thermodynamic critical angular frequency

Ωc =

ΔE , ΔL

(8.3)

where ΔE is the energy difference between a condensate with and without a vortex, and ΔL is the orbital angular momentum of the vortex state (non-vortex state has zero angular momentum). At zero temperature, as Ω is increased, each new vortex nucleated in the system corresponds to a quantum phase transition that changes the topology of the order parameter. In practice, the ideal rotating bucket is very difficult to realise in cold atom systems. Instead, when the asymmetric trapping potentials are rotated, vortices are often found to be nucleated only at much higher, Ω ℓ ≫ Ωc , angular frequencies:

⎛E ⎞ Ω ℓ = min ⎜ ℓ ⎟ , ⎝ ℓℏ ⎠

(8.4)

where Eℓ is the quasiparticle excitation energy of a Bogoliubov surface mode and ℓ is its orbital angular momentum quantum number, measured with respect to the condensate. Consequently, the vortex production in such experiments essentially relies on generation of quantum turbulence and its subsequent relaxation once the rotation of the potential stops. However, a true rotating bucket experiment with continuously rotating bucket walls could perhaps be achieved by interfering detuned high winding number LG laser modes that could provide a rotating hard-wall trapping potential for the condensate atoms with potentially very low heating rate.

8.5 Rotating thermal cloud FRS: rapidly rotating vortex lattices10 Rather than using external potentials to push the condensate atoms around to impart orbital angular momentum to them, it is possible to push the noncondensate atoms first to set them into rotation and allowing the condensate– noncondensate interactions to bring the condensate to a rotational equilibrium with the noncondensate. Removing the low angular momentum atoms from the vicinity of the rotation axis further increases the angular momentum per particle. This method has been employed to achieve the highest rotational rapidities constraining the condensate into the lowest Landau level in the mean-field quantum Hall regime.

10 V Schweikhard, I Coddington, P Engels, V P Mogendorff, and E A Cornell, Rapidly Rotating Bose–Einstein Condensates in and near the Lowest Landau Level, Physical Review Letters 92, 040404 (2004).

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8.6 Stirring FRS: laser stirring6,8, grid turbulence11,12 A BEC can be stirred by moving an obstacle, such as an off-resonant repulsive laser beam, that repels the superfluid particles within its volume. If the obstacle moves sufficiently rapidly with respect to the superfluid, comparable to the Landau or Feynman critical velocity, vortex nucleation may be achieved. Since only the relative velocity is important, it is possible to either move the stirrer keeping the condensate stationary in the laboratory frame of reference, or to keep the stirrer fixed in the laboratory frame and to move the whole condensate with respect to the stirrer. If the stirrer moves near along the boundary of the condensate, mostly one sign of vortices will be nucleated, where as if the moving obstacle lies well within the condensate, then topological defects will be nucleated as vortex–antivortex pairs conserving the total topological charge of the system. Multiple stirrers can be used to generate grid quantum turbulence.

8.7 Shaking bucket FRS: shaking condensate13,14 Similar to stirring a condensate, the bucket that confines the superfluid may also be shaken to transfer kinetic energy into the superfluid. In this case vortex–antivortex pairs are nucleated from the corrugations of the container (magnetic or optical potential confining the atoms) or by resonantly exciting collective modes of the condensate that may subsequently decay into quasiparticle excitations with orbital angular momenta.

8.8 Snaking instability FRS: soliton decay15

11 G Gauthier, M T Reeves, X Yu, A S Bradley, M Baker, T A Bell, H Rubinsztein-Dunlop, M J Davis, T W Neely, Giant vortex clusters in a two-dimensional quantum fluid Science, 364, 1264–7 (2019). 12 S P Johnstone, A J Groszek, P T Starkey, C J Billington, T P Simula and Kristian Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid Science 364, 1267–71 (2019). 13 E A L Henn, J A Seman, E R F Ramos, M Caracanhas, P Castilho, E P Olímpio, G Roati, D V Magalhes, K M F Magalhães, and V S Bagnato, Observation of vortex formation in an oscillating trapped Bose–Einstein condensate, Physical Review Letters 79, 043618 (2009). 14 N Navon, A L Gaunt, R P Smith, and Z Hadzibabic, Emergence of a turbulent cascade in a quantum gas, Nature 539, 72 (2016). 15 B P Anderson, P C Haljan, C A Regal, D L Feder, L A Collins, C W Clark, and E A Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose–Einstein Condensate, Physical Review Letters 86, 2926 (2001).

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Vortices may be generated as a decay product of other excitations such as solitons. The so-called snaking instability provides a decay channel for dark solitons. In this hydrodynamical instability the dark soliton first stretches into a sinusoidal shape and eventually breaks up, forming a chain of vortex–antivortex pairs.

8.9 Many-wave interference FRS: three wave interference16,17 Complex functions obey the linear superposition principle and it is possible to create phase singularities by superposing multiple smooth complex functions. A superposition of plane waves results in a complex valued function ψ (r) and may be expressed as N

ψ (r) = A(r)∑e i (kj ·r+ϕj ),

(8.5)

j=1

where A is the real valued amplitude, N is the number of interfering waves, kj are their wave vectors and ϕj are their global phases. The points rv at which the condition Re[ψ (r)] = Im[ψ (r)] = 0 is satisfied may host phase singularities corresponding to either a vortex or an antivortex. The superposition of two plane waves may produce a Young’s double-slit-type interference pattern (figure 8.1(a)). With an interference of three plane waves it is possible to produce interlaced vortex– antivortex honeycomb lattices (figure 8.1(b)). Four-wave interference yields rectangular vortex lattices (figure 8.1(c)) and by interfering five waves, vortex quasicrystals (figure 8.1(d)) can be produced.

8.10 Vortex–antivortex honeycomb lattices FRS: three wave interference18 The time-dependent superposition ψ (r, t ) state of three Gaussian wave packets

16

G Ruben and D M Paganin, Phase vortices from a Youngʼs three-pinhole interferometer, Physical Review E 75, 066613 (2007). 17 D R Scherer, C N Weiler, T W Neely, and B P Anderson, Vortex Formation by Merging of Multiple Trapped Bose–Einstein Condensates, Physical Review Letters, 98, 110402 (2007). 18 G Ruben, D M Paganin and M J Morgan, Vortex–lattice formation and melting in a nonrotating Bose– Einstein condensate, Physical Review A 78, 013631 (2008).

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Figure 8.1. Result of interfering (a) two, (b) three (c) four and (d) five plane waves whose wave vectors kj are distributed uniformly on the unit circle. The interference results in phase singularities marked with green (vortices) and white (antivortices) markers.

3

ψ (r , t ) =

⎛ −(Δp ℏ)2 ∣r − r ∣2

π Δp ℏ

∑ 1 + i(Δp)2 t mℏ exp ⎜ 2[1 + i(Δp)2 t mjℏ] ⎝

j=1

⎞ + iϕj ⎟ ⎠

(8.6)

can be expressed in terms of the centre-of-mass positions r j and momentum uncertainties Δp of the wave packets, and their global phases ϕj . The locations of the vortices and antivortices, expressed in polar coordinates (rv(t ), θv(t )), are

rv =

1 2

⎤2 1 ⎡ βN (n ) ⎛ βM (m) ⎞ r cos( ) ⎢ r − − − θ ⎜ ⎟ 3 2 3 ⎥ + 1 ⎝ sin2(θ ) ⎣ r2 ⎠ r3 ⎦

⎡ 1 ⎛ r − βN (n ) r ⎞ ⎤ 3 3 θv = arctan ⎢ ⎜ ⎟ − cos(θ3)⎥ , ⎣ sin(θ3) ⎝ r2 − βM (m) r2 ⎠ ⎦

(8.7)

(8.8)

where

β=

2 ⎡ (ℏt )2 + m 2(ℏ Δp)4 ⎤ ⎥, ⎢ ⎦ 3mℏ ⎣ t

and the functions

8-7

(8.9)

Quantised Vortices

M (m) = 2π [2 − l + 3(m − ϕ2 2π )] N (n ) = 2π [1 + l + 3(n − ϕ3 2π )]

(8.10)

are determined by integers (n, m ) with each pair corresponding to a location of one vortex (l = 0) or antivortex (l = 1). The vortices and antivortices are each arranged in triangular lattices interleaved in such a way that the resulting vortex structure is a honeycomb lattice of phase singularities as in the plane wave case (figure 8.1(b)).

8.11 Caustics and diffraction catastrophes FRS: focusing aberrated beams19,20,21, catastrophe theory22 Studies of focused light or matter waves have resulted in a hierarchy of discoveries and refinements to the fundamental theories of light and matter. In the ray picture the points where the rays cross are caustics (discovery) and formally correspond to infinite intensity (calling for refinement) of the rays. This unphysical result can be avoided by improving the theory and treating the focusing entities as waves, which allows the infinite intensity in the caustic regions to spread sufficiently to avoid the diverging intensities. Such refinement by the wave theory results in the prediction of the emergence of zeros (vortices) within the high-intensity caustic regions. However, the existence of a single point in space whose location could be determined to arbitrary precision violates the Heisenberg uncertainty principle. Further refinement by quantum field theory shows that quantum fluctuations partially fill up the vortex cores, softening the phase singularities, and leads to the result that it is not possible to determine the exact position of the vortex phase singularity more accurately than the size of the vortex core. Once the quantum field theory becomes superseded by a better theory, perhaps some form of string theory, the fine structure within the vortex core inside a caustic inside a ray, should be a fruitful place to look to test the next level of theoretical refinement. As illustrated in figure 8.2, zooming inside a caustic region reveals a topologically robust structure composed of vortices and antivortices that emerges due to the interference of many waves. The structure of such diffraction catastrophes are classified by the catastrophe or singularity theory. The two most common potential functions characterising, respectively, the fold and cusp catastrophes are

19

M V Berry, Waves and Thomʼs theorem, Advances in Physics 25, 1 (1975). T P Simula, T C Petersen, and D M Paganin, Diffraction catastrophes threaded by quantized vortex skeletons caused by atom-optical aberrations induced in trapped Bose–Einstein condensates, Physical Review A 88, 043626 (2013). 21 T C Petersen, M Weyland, D M Paganin, T P Simula, S A Eastwood, and M J Morgan, Electron Vortex Production and Control Using Aberration Induced Diffraction Catastrophes, Physical Review Letters 110, 033901 (2013). 22 V I Arnold, Catastrophe Theory, Springer-Verlag (1992). 20

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Figure 8.2. Focusing aberrated matter waves. (i) Bose–Einstein condensate (BEC) in ground state of a ring trap. (ii) Applying a perturbation to the trapping potential imprints a phase profile to the condensate wave function affecting its momentum distribution. (iii) The caustic, diffraction catastrophe, and the quantized vortex skeleton emerge during the time of flight due to multiwave interference.

Vfold = x 3 + ax ;

and

Vcusp = x 4 + ax 2 + bx ,

(8.11)

with the control parameters a and b determining the structure of the stable and unstable solutions.

8.12 Vortex quasicrystals FRS: quasicrystals23,24 Quasicrystals are structures which may have, for instance, 10-fold discrete rotational symmetry and lack translational symmetry. Similarly, vortex quasicrystals are quasiperiodic arrangements of vortices. Despite them lacking translational invariance, quasicrystalline structures can be space-filling, as can be demonstrated using Penrose tilings. Figure 8.3 illustrates a Fibonacci quasicrystal structure with the corresponding vortex quasicrystal that emerges within the structure due to five-wave interference. Due to the absence of diagonal long-range ordering, phonons cannot propagate through quasicrystals. Instead of sound waves, the elementary excitations in quasicrystals are called phasons, which correspond to spatial rearrangements of the constituents of the quasicrystal.

8.13 Vortex phasons FRS: phason25

23 D Shechtman, I Blech, D Gratias, and J W Cahn, Metallic Phase with Long-Range Orientational Order and No Translational Symmetry, Physical Review Letters 53, 1951 (1984). 24 D Levine and P J Steinhardt, Quasicrystals: A New Class of Ordered Structures, Physical Review Letters 53, 2477 (1984). 25 P Bak, Phenomenological Theory of Icosahedral Incommensurate (‘Quasiperiodic’) Order in Mn–Al Alloys, Physical Review Letters 54 1517 (1985).

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Figure 8.3. Fibonacci vortex quasicrystals. (a) Diffraction peaks in momentum space related to an icosahedral quasicrystal. Interference of the five green spots results in the quasiperiodic structure shown in (b). The circle in (a) is shown to highlight the equal magnitude of the five momentum components and the other spots at larger momenta illustrate higher order diffraction peaks. The magnitudes ∣k1∣ and ∣k2∣ of the two momentum peaks denoted by interval markers form the golden ratio ∣k2∣/∣k1∣ = (1 + 5 )/2 . The four momentum components within the box with dashed boundary in (a) result in the four-wave interference pattern in (c) where the fringe spacing corresponds to the Fibonacci sequence. Moving one of the diffraction peaks as indicated in (a) results in phason strain, illustrated in (d). In (b) the vortices (green markers) and antivortices (white markers) are also arranged in a quasiperiodic pattern.

A phason is a quasiparticle excitation of a quasicrystal. In crystals, phonons that are propagating sound waves are an emergent collective degree of freedom due to the underlying periodic arrangements of the particles. In a quasicrystal, long-range periodic order does not exist and therefore phonons cannot exist either. The elementary excitations that replace phonons in quasicrystals are called phasons and may be viewed as phonons of a higher dimensional crystalline pre-image, projected to the lower dimensional quasicrystal space. The presence of a phason strain in a quasicrystal can be detected by analysing the Fourier components of the quasicrystalline structure, as illustrated in figure 8.3. Similarly, in a vortex quasicrystal, vortex phasons can emerge which replace the Tkachenko sound modes of Abrikosov vortex lattices.

8.14 Vortex Moiré superlattices ´ patterns26 FRS: Moire The Moiré patterns are an emergent effect where a new lattice structure, absent in the elementary constituents, is produced by overlaying two or more regular semitransparent patterns. For example, by overlaying two triangular lattice,s as shown in figure 8.4(a), that are rotated with respect to each other a certain angle θ, as in figure 8.4(b), will result in new structures shown in figure 8.4(c)–(e). Such emergent structures are a collective property of its constituent structures. Similarly, by interfering complex valued waves it is possible to produce Moiré vortex patterns. Examples of such are shown in figure 8.4(f)–(h) where two sets of waves producing three-wave interference are rotated relative to each other, producing a six-wave 26

I Amidror, The Theory of the Moiré Phenomenon, Springer (2009).

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Figure 8.4. Moiré superlattices produced by overlapping sets of waves. (a) shows the probability density ∣ψ3(r)∣2 produced by an interference of three plane waves as in figure 8.1(b). (b) shows the Moiré pattern that results from overlaying two copies of the pattern shown in (a) and rotated by a small angle θ with respect to each other. (c)–(e) show the probability density ∣ψ3(r, 0)∣2 + ∣ψ3(r, θ )∣2 that results from overlaying two probability densities, each of which is the result of an interference of three plane waves as in (b), for θ = 2π /8, 2π /13, and 2π /35 radians, respectively. (f)–(h) show the probability density ∣ψ3(r, 0) + ψ3(r, θ )∣2 produced by an interference of six plane waves for θ = 2π /8, 2π /13, and 2π /35 radians, respectively. The locations of vortices and antivortices are denoted by green and white markers, respectively.

interference pattern with the locations of vortices and antivortices denoted by green and white markers, respectively.

8.15 Synthetic gauge fields FRS: synthetic magnetic field technique27, synthetic spin-orbit coupling28, artificial gauge potentials29

27 Y -J Lin, R L Compton, K Jiménez-García, J V Porto, and I B Spielman, Synthetic magnetic fields for ultracold neutral atoms, Nature 462, 628 (2009). 28 Y -J Lin, K Jiménez-García, and I B Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471, 83 (2009). 29 J Dalibard, F Gerbier, G Juzeliūnas and P Öhberg, Colloquium: Artificial gauge potentials for neutral atoms, Reviews of Modern Physics 83, 1523 (2011).

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The effect of a rotating bucket on neutral atoms may be quantified by the term −ΩLz in the Hamiltonian

HΩ =

p2 − ΩL z , 2m

(8.12)

in the rotating frame. This may be equivalently expressed as

HΩ =

(p − q A)2 1 − mΩ2r 2 , 2m 2

(8.13)

with the choice q A = mΩ × r . This shows that, similarly to how in quantum mechanics electromagnetic field couples to an electric charge via the vector potential, the effect of rotation on neutral atoms can be expressed in terms of a vector potential with the rotation resulting in a (synthetic magnetic) gauge field. For sufficiently strong synthetic magnetic field (rotation), quantised vortices are nucleated in the superfluid. However, the rotation is accompanied by a scalar potential that causes centrifugal expansion of the condensate. In a spinor BEC, the spin degree of freedom makes it possible to realise synthetic magnetic fields while avoiding the scalar potential by generating the vector potential only. The essential ingredient is to use external (real) electromagnetic fields to impart linear momentum on the condensate atoms in a spin-state-dependent way. The same basic principle can also be used to realise various forms of synthetic spin–orbit L · S couplings for neutral superfluid atoms.

8.16 Optical flux lattices FRS: optical flux lattices30,31, charge-flux quantum liquids32 Conventional optical lattices may be created by interfering counter-propagating laser beams to yield scalar optical lattice potentials such as

VOL(x , y ) = V0[cos(kx )2 + cos(ky )2 + 2ϵ1 · ϵ2 cos(ϕ) cos(kx ) cos(ky )],

(8.14)

where the amplitude V0 can be controlled using the power of the laser, k = 2π /λ is the wave vector of light and ϕ and ϵi are the relative phase and polarisation vectors of the two beams. Such scalar potentials couple to a BEC via the dipole moment d of the atoms through the usual term U = −d · E, where E is the electric field, resulting in an additional scalar potential term in the Hamiltonian HOL = H0 + VOL . However, by deploying many-wave interference, it is also possible to create vector optical flux lattices whose phase structure is non-trivial and essentially contain optical vortices at the lattice sites. These are called optical flux lattices and they can 30

N R Cooper, Optical Flux Lattices for Ultracold Atomic Gases, Physical Review Letters 106, 175301 (2011). G Juzeliūnas and I B Spielman, Flux lattices reformulated, New Journal of Physics 14, 123022 (2012). 32 T Simula, Synthetic charge–flux quantum liquids, Physical Review B 85, 144521 (2012). 31

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be coherently coupled to the internal states of the atoms to yield very high flux density per atom to engineer various topological phases of matter. The flux lattice potentials such as

VFL(x , y ) = V0[cos(kx )eˆ x + cos(ky )eˆ y + sin(kx ) cos(ky )eˆ z]

(8.15)

couple to a BEC via the magnetic moment μ of the atoms through the usual term U = −μ · B, where B is the magnetic field, resulting in an additional term in the Hamiltonian HFL = H0 + VFL · F, where F is a vector of Pauli spin matrices.

8.17 Filtered speckle fields FRS: speckle fields33, coarsening dynamics34,35 Consider a complex valued function Ψ(r) with vanishing spatial phase coherence. Such a state can be simply generated by drawing a set of pairs of random numbers {ai , bi } ∈ [−π , π ] from a uniform distribution and setting these to be equal to the real Re[arg(Ψ(ri ))] = ai and imaginary Im[arg(Ψ(ri ))] = bi components of the complex phase of the function Ψ(r) at each point ri . Using low-pass filtering to cut out high-spatial frequencies of the phase function S (r) = arg[Ψ(r)] results in phasecoherent patches with vortices and antivortices at the boundaries of such coherent patches; see figure 8.5. Such Fourier filtering process could qualitatively represent the outcome of dynamical evolution corresponding to a great variety of physical scenarios: • In a decaying two-dimensional (2D) quantum turbulence the evaporative heating mechanism due to vortex–antivortex annihilation leads to the vortex number decay and increasingly smoother phase structure of the condensate wave function.

Figure 8.5. Fourier-filtered speckle fields S (r) obtained by filtering away successively longer wavelength Fourier modes (a)–(c). This results in ever smoother function, increasing phase coherence length, and fewer number of (vortices) topological defects. The phase singularities denoted with green and white markers correspond to vortices and antivortices, respectively.

33

D M Paganin, Coherent X-Ray Optics, Oxford University Press (2006). A J Bray, Theory of phase-ordering kinetics, Advances in Physics 43, 357 (1994). 35 L A Williamson and P B Blakie, universal Coarsening Dynamics of a Quenched Ferromagnetic Spin-1 Condensate, Physical Review Letters 116, 025301 (2016). 34

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• In a driven 2D quantum turbulence an inverse energy cascade does not remove the short wavelengths but rather increases the weight of the longer wavelength modes, resulting in similar growth of the phase coherence. • In dissipative systems, removal of high-energy particles or waves from the system is analogous to such low-pass Fourier filtering of the excitation spectrum. An evaporative cooling of atoms used in the production of ultracold atomic gases and BECs is a typical example of such process. • Cooling through a phase transition, such as the Berezinskii–Kosterlitz– Thouless transition, or after a rapid Kibble–Zurek type quench often results in qualitatively similar decimation of topological defects and post-quench coarsening dynamics. • Propagation of incoherent electromagnetic fields through a waveguide or defocusing a partially coherent optical field in free space are examples where the spatial phase coherence of the underlying complex valued field may increase as a function of time.

8.18 Kibble–Zurek mechanism and quenches FRS: Kibble-Zurek mechanism36,37, laboratory experiments38,39,40 Quantized vortices, as well as other topological defects, may also be produced by quenching the system across a phase transition by varying some externally controllable parameter such as the temperature. The quench does not affect the total internal energy of the system but after the quench the system’s ground state may have a significantly lower energy. The excess energy, the difference between the total energy and the new ground state energy, is then shed off into topological defects such as quantised vortices. This process is known as the Kibble–Zurek mechanism and is, for instance, predicted to have resulted in the formation of cosmic strings in the early universe. If a BEC phase transition is crossed sufficiently rapidly, the phase coherence of the condensate will begin to grow independently in causally separated patches a certain distance apart. Topological defects will then be nucleated at the intersections of such locally phase-coherent patches when they merge.

36 T W B Kibble, Topology of cosmic domains and strings, Journal of Physics A: Mathematical and General 9, 1387 (1976). 37 W H Zurek, Cosmological experiments in superfluid helium?, Nature 317, 505 (1985). 38 C Bäuerle, Y M Bunkov, S N Fisher, H Godfrin and G R Pickett, Laboratory simulation of cosmic string formation in the early Universe using superfluid 3He, Nature 382, 332 (1996). 39 V M H Ruutu, V B Eltsov, A J Gill, T W B Kibble, M Krusius, Y G Makhlin, B Plaais, G E Volovik and W Xu, Vortex formation in neutron-irradiated superfluid 3He as an analogue of cosmological defect formation, Nature 382, 334 (1996). 40 C N Weiler, T W Neely, D R Scherer, A S Bradley, M J Davis and B P Anderson, Spontaneous vortices in the formation of Bose–Einstein condensates, Nature 455, 948 (2008).

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Generically, the behaviour of systems in the vicinity of the critical scaled temperature t˜ = ∣1 − T /Tc∣ of second-order phase transitions can be classified into universality classes by their critical exponents including z and ν that determine scaling laws for observables such as relaxation time τ = τ0t˜ −νz and healing length ξ = ξ0t˜ −ν . These may be used to obtain the propagation speed of the sonic horizon v = ξ /τ = v0t˜ ν(z−1). The freeze out time 1

τˆ = (τ0τQνz )1+νz ,

(8.16)

expressed in terms of the quench time τQ that controls the speed of the temperature change in the quench, determines the moment at which the system’s reflexes are no longer able to adiabatically relax to the changing environment. For later times, coherence grows in local patches independently, which seeds defect formation when such patches come into contact with each other. The inter-defect distance can be obtained as ν

⎛ τQ ⎞1+νz ξˆ = ξ0⎜ ⎟ ⎝ τ0 ⎠

(8.17)

d −d and it determines the initial defect density. One defect per a domain of size ξˆ s d , where ds is the dimension of the space and dd is the dimension of the defect, will be generated. Hence a quench of a bulk superfluid, ds = 3, is expected to result in an −2 initial vortex line, dd = 1, density ∝ξˆ , which will subsequently decay via various nonlinear processes.

8.19 Berezinskii–Kosterlitz–Thouless mechanism FRS: BKT mechanism41,42,43, experimental observations44,45,46,47, thermal vortex-antivortex pair production in Bose gas48

41 J M Kosterlitz, Nobel Lecture: Topological defects and phase transitions, Reviews of Modern Physics 89, 040501 (2017). 42 V L Berezinskiıˇ Destruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems, Soviet Journal of Experimental and Theoretical Physics 34, 610 (1972). 43 J M Kosterlitz and D J Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics 6, 1181 (1973). 44 D J Bishop and J D Reppy, Study of the Superfluid Transition in Two-Dimensional 4He Films, Physical Review Letters 40, 1727 (1978). 45 Z Hadzibabic, P Krüger, M Cheneau, B Battelier and J Dalibard, Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas, Nature 441, 1118 (2006). 46 P Cladé C Ryu, A Ramanathan, K Helmerson, and W D Phillips, Observation of a 2D Bose Gas: From Thermal to Quasicondensate to Superfluid, Physical Review Letters 102, 170401 (2009). 47 C -L Hung, X Zhang, N Gemelke and C Chin, Observation of scale invariance and universality in twodimensional Bose gases, Nature 470, 236 (2011). 48 T P Simula and P B Blakie, Thermal Activation of Vortex–Antivortex Pairs in Quasi-Two-Dimensional Bose– Einstein Condensates, Physical Review Letters 96, 020404 (2006).

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Vortices may be produced spontaneously from thermal energy available in the system via the Berezinskii–Kosterlitz–Thouless (BKT) mechanism. At sufficiently high temperature the Helmholtz free energy F = U − TS of the system favours spontaneous nucleation of vortex–antivortex pairs in the system, which greatly increases the entropy for only a modest cost of internal energy. At low temperatures the vortices and antivortices preferentially form closely bound pairs and hence the BKT transition with a critical temperature Tc is also known as a topological vortex binding–unbinding transition. In terms of the Ehrenfest classification, the transition is of infinite order and does not involve continuous symmetry breaking. In 2D uniform systems, in the thermodynamic limit, true long-range order (and therefore Bose–Einstein condensation) cannot persist, yet this does not rule out superfluidity that can be achieved with correlations that only possess quasi-longrange ordering. On the high-temperature side T > Tc of the BKT transition two-point phase correlations G (r˜ ) = e i〈ϕ(r˜ )ϕ(0)〉, with r˜ = ∣r − r′∣/r0 , of the phase ϕ decay exponentially

G (r˜ ) ∝ e−αr˜ ,

(8.18)

with a decay constant α. On the superfluid side T < Tc of the transition two-point phase correlations decay algebraically

G (r˜ ) ∝ r˜ −λ(T ),

(8.19)

where the correlation length

λ(Tc ) =

m 2kBTc 1 = 2 2π ℏ ρs (Tc ) 4

(8.20)

determines the Nelson–Kosterlitz jump in the superfluid density at the critical point. Note that the superfluid density is not equal to the condensate density. The critical temperature

Tc =

κ 2σs ℏωk m 0 = 8πkB 2kB m

(8.21)

where σs = ρs d is the areal superfluid density, ρs the three-dimensional bulk superfluid density, d the thickness of the quasi-2D fluid, ωk is the kelvon frequency and m0 is the inertial kelvon based mass of a vortex. The critical temperature may be obtained by considering vanishing of the free energy cost ΔF = ΔE − T ΔS to create (i) a single vortex in a superfluid or (ii) a single vortex–antivortex pair (two vortices) in a superfluid. In both cases the entropy gained scales as S ∝ ln rc /R , where rc is the vortex core size and R is the system size. However, the energy of an isolated vortex E1 ∝ ln R /rc where as that of a vortex–antivortex pair E2 = 2E1, where R is now interpreted as the distance between the vortex and the antivortex. This means that while (i) yields the correct BKT critical temperature T1 = Tc , (ii) yields a critical temperature T2 = 2Tc –twice the BKT critical temperature. It turns out that while (i) indeed yields the correct BKT critical temperature, (ii) yields the correct critical temperature for the Hauge–Hemmer pair collapse transition of a zero-core 2D 8-16

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Coulomb gas. Introducing a hard-core to the Coulomb gas model shifts the critical temperature and by choosing such a vortex core size to match the temperature dependent self-consistent vortex core size in the BKT theory, the result (i) can be recovered. Strictly, it is not possible to create a single isolated vortex because it would violate topological charge conservation and as such the free energy in (i) must be interpreted in terms of a collective effect, including screening, due to all vortices.

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IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 9 Topological quantum computation

The basic idea underlying topological quantum computation relies on topological charge conservation that prevents certain isolated quasiparticles such as quantised vortices in superfluids from being destroyed by any local operation without supplying an infinite amount of energy to the system, where infinite means comparable to the total energy of the pertinent system. Such topological protection is also thought to underpin an inherent resilience of topological qubits toward the usual types of environment-induced decoherence that the conventional qubits are very vulnerable to. To be useful for quantum computation, such topologically protected (quasi)particles must also be governed by exotic fusion rules specific to non-abelian anyons that are peculiar to two-dimensional (2D) systems.

9.1 Non-abelian anyons FRS: quantum computation with non-abelian anyons1,2 A non-abelian anyon model has a trivial (vacuum) anyon, here denoted by 1, and a set of non-trivial anyons α. At least one of the non-trivial anyons must be nonabelian, meaning that it may fuse with an another anyon in more than one way. The notation α ⊗ α = 1 ⊕ β ⊕ γ means that when α anyon fuses with α anyon the outcome may be either 1, β or γ anyon. Generically, topological quantum computation with non-abelian anyons consists of three elementary operations: • creation of an array of anyons with each logical topological qubit comprising several anyons;

1 2

M H Freedman, P/NP, and the Quantum Field Computer, PNAS 95, 98 (1998). A Y Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).

doi:10.1088/2053-2571/aafb9dch9

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• braiding the world lines of the anyons by moving them around in the 2D physical space to implement unitary gate operations acting in the Hilbert space spanned by the fusion paths of the topological qubits; and • fusion (merging) of the anyons to measure the final state of the topological qubits and thereby the outcome of the computation.

9.2 Topological qubits FRS: topological qubit3,4 In the Fibonacci anyon model, a single logical topological qubit may be constructed, for example from four Fibonacci anyons in a row, labelled from one to four. The fourth, inert, anyon does not participate in the computation and is only required for total ‘charge’ conservation to guarantee that, for example, if all of the four anyons are fused then they all become annihilated, leaving only vacuum state left. Consequently, a topological qubit could essentially be constructed from the first three anyons only. If the anyons one and two annihilate when fused, the topological qubit is in a state ∣0〉. If the anyons 1 and 2 do not annihilate when fused, the topological qubit is in a state ∣1〉. It is the non-abelian topological structure of such anyons that enables superposition states in which anyons 1 and 2 have simultaneously a non-zero probability to annihilate or not to annihilate. More specifically, braiding the second and third anyon can change the fusion outcome (annihilation probability) of the first two anyons, hence placing the topological qubit in a superposition of states ∣0〉 and ∣1〉. A many-qubit state ∣Ψ〉 = ∣01011…〉 = (∣0〉 ⊗ ∣1〉 ⊗ ∣0〉 ⊗ ∣1〉 ⊗ ∣1〉…) that corresponds to a bit string determined by the states of the individual topological qubits can be mapped onto a unique sequence of anyon fusion outcomes. Operationally, braiding the anyons corresponds to a change of basis and is determined by the unitary operators called R and F moves. The F -matrix element F(αβγδ )θη and the R -matrix element R αβ γ can be represented graphically as shown in figure 9.1. The R and F -matrices vary from anyon model to anyon model and may be determined using consistency rules known as the pentagon and hexagon relationships.

9.3 Quantum dimension FRS: quantum dimension5 The quantum dimension dα of an anyon α is determined by the relation 3

C Nayak, S H Simon, A Stern, M Freedman and S Das Sarma, Non-Abelian anyons and topological quantum computation, Reviews of Modern Physics 80, 1083 (2008). 4 B Field and T Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 3, 045004 (2018). 5 J Preskill, Lecture Notes for Physics 219: Quantum Computation, California Institute of Technology (2004).

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Figure 9.1. Diagrammatic representation of the F and R moves effecting a change of basis in the topological qubit system.

d αdβ =

∑Nαβγd γ ,

(9.1)

γ γ where the integer Nαβ counts the number of distinguishable ways anyons α and β may fuse to yield anyon γ. The total quantum dimension of the anyon model is

D2 =

∑d α2.

(9.2)

α

Physical interpretations of the quantum dimension dα include the following: • Create two anyon–antianyon pairs (a row of four anyons) from the vacuum. The inverse of the square of the quantum dimension is the probability, p = 1/d 2 , that the two anyons (2 and 3) in the middle annihilate when fused (merged). • Let P (n ) be the number of distinct fusion paths for n anyons, each of which corresponds to one state in the anyons’ Hilbert space. The quantum dimension of anyon α is the ratio, limn→∞P (n + 1)/P (n ) = dα , in the ‘thermodynamic limit’ of infinitely many anyons—the rate at which the size of the anyons’ Hilbert space grows when another anyon of the same type is added in the system. • The quantum dimension is a Perron–Frobenius eigenvalue of the matrix Nα that defines how anyon α fuses with other anyons. For anyons corresponding to representations of underlying finite groups, their quantum dimensions are integers and are equal to the dimensions of their representations. Particles whose quantum dimension d = 1 are abelian anyons. Particles whose quantum dimension d > 1 are non-abelian anyons. Non-abelian anyons are thought to be capable of topologically protected universal quantum computation if the square of their quantum dimension is not an integer. If the square of the quantum dimension of a non-abelian anyon model is an integer it does not provide a universal gate set for topological quantum computation by braiding alone but could still be capable of universal quantum computation if complemented by non-topological gate operations.

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9.4 Majorana Ising anyon model FRS: Ising anyons6,7,8 Majorana quasiparticle zero modes are a prominent candidate for the realisation of an Ising anyon model. The fusion rules of the Ising anyon model with three anyon types, 1, σ and ψ, are

σ⊗σ=1⊕ψ

(9.3)

ψ⊗ψ=1

(9.4)

ψ⊗σ=σ

(9.5)

1 ⊗ α = α,

(9.6)

where α ∈ {1, σ , ψ }. The R -matrix elements and F -matrix elements that define the braiding statistics of the non-abelian σ anyons are determined to be i 3π 8 iπ 2 R1σσ = e−iπ 8; R σσ ; R1ψψ = e iπ ; R σψ ; ψ = e σ = e

F(σσσσ )11 =

(9.7)

1 1 1 1 . (9.8) ; F(σσσσ )ψψ = − ; F(σσσσ )1ψ = ; F(σσσσ )1ψ = 2 2 2 2

The quantum dimension dσ = 2 and the elementary braid matrices σ1 and σ2 for the three-strand Artin braid group B3 for the Ising anyon model are

⎛ ⎞ e−iπ 8 ⎛ e iπ 4 e−iπ 4 ⎞ σ1 = e−iπ 8⎜1 1⎟ ; σ2 = ⎜ ⎟. ⎝1 i ⎠ 2 ⎝ e−iπ 4 e iπ 4 ⎠

(9.9)

9.5 Fibonacci anyon model FRS: Fibonacci anyons9,10

6 S Das Sarma, M Freedman and C Nayak, Majorana zero modes and topological quantum computation 1, 15001 (2015). 7 C Nayak, S H Simon, A Stern, M Freedman and S Das Sarma, Non-Abelian anyons and topological quantum computation, Reviews of Modern Physics 80, 1083 (2008). 8 J Alicea and A Stern, Designer non-Abelian anyon platforms: from Majorana to Fibonacci, Physica Scripta 2015, 014006 (2015). 9 S Trebst, M Troyer, Z Wang, A W W Ludwig, A Short Introduction to Fibonacci Anyon Models, Progress of Theoretical Physics Supplement 176, 384 (2008). 10 B Field and T Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 3, 045004 (2018).

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The fusion rules of the Fibonacci anyon model with two anyon types, 1 and τ, are

τ⊗τ=1⊕τ

(9.10)

1 ⊗ α = α,

(9.11)

where α ∈ {1, τ}. The R -matrix elements and F -matrix elements that define the braiding statistics of the non-abelian τ anyons are determined by

R1ττ = e−i 4π 5; F(ττττ )11 =

i 3π 5 R ττ ; τ = e

1 1 1 1 ; F(ττττ )1τ = ; F(ττττ )1τ = ; F(ττττ )ττ = − . ϕ ϕ ϕ ϕ

(9.12) (9.13)

The quantum dimension dτ of the Fibonacci anyon equals the golden ratio ϕ = d τ = (1 + 5 )/2 and gives the Fibonacci anyon its name. The elementary braid matrices σ1 and σ2 for the three-strand Artin braid group B3 for the Fibonacci anyon model are

⎛ e − i 4π 5 0 ⎞ σ1 = ⎜ ⎟; i 3π 5 ⎠ ⎝0 e

⎛ e i 4π 5 e − i 3π 5 ⎞ ⎜ ⎟ ϕ ⎟ ⎜ ϕ . σ2 = ⎜ − i 3 π 5 1 ⎟⎟ ⎜⎜ e − ϕ ⎟⎠ ⎝ ϕ

(9.14)

9.6 Model k anyons FRS: Model k anyons11,12 The Kac–Moody algebras SU(2) ℓ generate a countably infinite set of anyon models classified by an integer ℓ = k − 2. Such Model k anyons are employed in the Aharonov–Landau–Jones topological quantum algorithm and may also be used for solving the Jones polynomial of knots and links at the kth root of unity exactly. The quantum dimensions of Model k anyons, labelled by j = 0, 1 2, 2 2, …, (k − 2) 2, are dkj = sin((2j + 1)π /k )/sin(π /k ). For k = 4, the non-abelian Model k anyon j = 1 2 reduces to the Ising anyon and for k = 5, the non-abelian Model k anyon j = 1 reduces to the Fibonacci anyon. For d∞ = 2 ordinary SU(2) spins are recovered. Only the cases k = 5 and k = 7 are thought to yield universal gate set by braiding alone.

11

B Field and T Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 3, 045004 (2018). 12 Gils et al Physical Review B 87, 235120 (2013)

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9.7 Non-abelian vortex anyons FRS: Non-abelian vortex anyons13 For a spin-2 Bose–Einstein condensate (BEC), the fluxons (vortex anyon types) of the cyclic-tetrahedral phase are classified by seven conjugacy classes:

(I) {(η ,  )} (II) {(η , − )} (III) {(η , iσν ), (η , −iσν )} (IV) {(η + 1 3, σ˜ ), (η + 1 3, −iσνσ˜ )} (V) {(η + 1 3, −σ˜ ), (η + 1 3, iσνσ˜ )} (VI) {(η + 2 3, −σ˜ 2 ), (η + 2 3, −iσνσ˜ 2 )} (VII) {(η + 2 3, σ˜ 2 ), (η + 2 3, iσνσ˜ 2 )},

(9.15)

and similarly for those of the biaxial nematic phase:

(I) {(η ,  )} (II) {(η , − )} (III) {(η , ±iσx ), (η , ±iσy )} (IV) {(η , iσz ), (η , −iσz )} (V) {(η + 1 2, σ˜ ), (η + 1 2, −iσzσ˜ )} (VI) {(η + 1 2, −σ˜ ), (η + 1 2, iσzσ˜ )} (VII) {(η + 1 2, ±iσxσ˜ ), (η + 1 2, ±iσyσ˜ )},

(9.16)

where σν , for ν = x , y, z are spin-1/2 Pauli matrices and σ˜ ≡ ( + iσx + iσy + iσz )/2. Each specific vortex flux may be represented using a shorthand notation, ±X νη ≡ (η + aX , gXν ). The X is a roman numeral corresponding to the class number, the subscript η is the winding number of the U(1) rotation, and aX is a class-specific constant. The superscript ν specifies the axis of the σν Pauli matrix generator of the classspecific SU(2) rotation gXν , and the sign of gXν corresponds to the sign of the vortex label. The composition rule

(±X αη)(±Y νβ) = (η + a X + ν + a Y, gXαgYβ )

(9.17)

determines the fusion outcome of a pair of vortex fluxes and can be used for identifying type-I and type-II non-abelian vortex pairs. For a pair of type-I nonabelian fluxons their fusion may have multiple outcomes but the initial vortices and the braided vortices belong to the same conjugacy class. The consequence of this is that topology does not enforce rung formation upon collision of type-I non-abelian 13 T Mawson, T Petersen, and T Simula, Braiding and fusion of non-Abelian vortex anyons, arXiv:1805.10009, (2018).

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vortices such that their collision dynamics is abelian. For a pair of type-II nonabelian fluxons their fusion may have multiple outcomes but braiding two such vortices may result in the braided vortices to belong to a different conjugacy class than the original vortices. For type-II non-abelian vortices rung formation is topologically enforced upon their collision such that not only their fusion rule but also their collision dynamics are non-abelian. Generically, the non-abelian vortex anyon models are defined by the quantum double of finite quantum groups with the full set of particles comprising fluxons, chargeons and dyons.

9.8 Annihilation, pass-through and rungihilation FRS: collision dynamics of non-abelian vortices14,15 If the topological invariant of a combination A · B of vortices A and B is equivalent to a trivial vacuum defect, the two vortices may annihilate. For example, following the composition rule equation (9.17), in a scalar BEC a fusion of a vortex A = +1 and an antivortex B = −1 yield A · B = 0, which is a vortex-free state. Similarly, in a spin-2 BEC in the cyclic phase A = {+1, iσx} and B = {−1, −iσx} would result in a trivial vacuum defect A · B = {0,  }. These are purely topological considerations and what happens in a practical setting depends on the details of the energetics and dynamics. If the total topological charge of two defects is non-trivial the defects may pass through each other upon colliding. For example, a fusion of two scalar vortices A = +1 and B = +1 result in A · B = +2, which is a doubly quantized vortex. Topologically, the two vortices could therefore simply pass through each other. However, the large Coulomb-like repulsion between two same-sign vortices renders it energetically unlikely for them to be brought in the proximity of each other by the dynamics. For certain vortices in spinor BECs such strong repulsive energy barriers do not exist and two vortices may pass through each other without apparent interaction. Moreover, if the topological charges of two vortices A and B are noncommutative such that A · B ≠ B · A, they may be topologically forced to rungihilate, in which two non-abelian vortices fuse to form a rung vortex defect. Just like pass-through, rungihilation is also topologically allowed to occur also for abelian vortex pairs. However, just like overlapping two same-sign scalar vortices is improbable due to energetic reasons, pass-through and rungihilation are typically not observed for abelian vortices. In contrast, for certain non-abelian vortex pairs, rungihilation is the only topologically allowed outcome when two vortices collide, and it is in this sense that the non-abelian algebra of such vortices dramatically alters their dynamics.

14 M Kobayashi, Y Kawaguchi, M Nitta, and M Ueda, Collision Dynamics and Rung Formation of nonAbelian Vortices, Physical Review Letters 103, 115301 (2009). 15 T Mawson, T C Petersen, and T Simula, Collision dynamics of two-dimensional non-Abelian vortices, Physical Review A 96, 033623 (2017).

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9.9 Non-abelian vortex anyon models FRS: non-abelian vortex anyons16, quantum double17 Most (F > 1) spinor BECs possess ground states with order parameters whose vortices are classified by non-abelian finite symmetry groups, which facilitate a great variety of non-abelian anyon models. All such anyons have integer quantum dimensions that derive from the dimensions of the underlying finite group representations. The simplest spinor BEC capable of hosting non-abelian vortex anyons is a spin-2 BEC, which has two ground state phases, cyclic-tetrahedral and biaxial nematic, whose topological defects are described by non-abelian symmetry groups. For the vortices in the cyclic-tetrahedral ground state the relevant group is the 24element binary tetrahedral symmetry group T *, which is isomorphic to SL2(3). For the vortices in the biaxial nematic ground state the relevant group is the 16-element non-abelian binary dihedral-4 group D4*. The full structure of the resulting anyon models is determined by the quantum double D(H ). For the cyclic-tetrahedral phase, the H-charges correspond to irreducible representations of the centralizer groups of T *. The centralizers for the fluxon types are T * (I, II), Z4 (III), and Z6 (IV - VII) with 7, 4, and 6 irreducible representations, respectively. In total, for a given η, the cyclic-tetrahedral phase anyon system has one vacuum state and 41 nontrivial particles comprising 6 fluxons, 6 chargeons, and 29 dyons. For the biaxial nematic phase, the H-charges correspond to irreducible representations of the centralizer groups of D4*. The centralizers are D4* (I, II), Z4 (III, VII), and Z8 (IV - VI) with 7, 4, and 8 irreducible representations, respectively. In total, for a given η, the biaxial nematic phase anyon system has one vacuum state and 45 non-trivial particles comprising 6 fluxons, 6 chargeons, and 33 dyons.

9.10 Vortex anyon creation, pinning, braiding, and fusion FRS: optical control of vortex anyons18,19 Vortex anyons can be produced experimentally by ‘de-fusing’ them from the BEC vacuum pairwise. Such vortex pair-creation conserves the total topological winding number and can be achieved analogously to the way it is possible to create ‘wing-tip’ vortex–antivortex pairs in classical fluids by moving an obstacle, such as a spoon or 16 T Mawson, T Petersen, and T Simula, Braiding and fusion of non-Abelian vortex anyons, arXiv:1805.10009, (2018). 17 M de Wild Propitius and F A Bais, Discrete gauge theories, in G Semenoff and L Vinet Eds, Particles and Fields, CRM Series in Mathematical Physics, Springer (1998). 18 A Rakonjac, A B Deb, S Hoinka, D Hudson, B J Sawyer, and N Kjærgaard, Laser based accelerator for ultracold atoms, Optics Letters 37, 1085 (2012). 19 E C Samson, K E Wilson, Z L Newman and B P Anderson, Deterministic creation, pinning, and manipulation of quantized vortices in a Bose–Einstein condensate, Physical Review A 93, 023603 (2016).

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an aircraft, through the fluid. Alternatively, one could deploy sophisticated phaseimprinting methods using, for example, Laguerre–Gauss laser modes to create the vortices. Once the vortices have been nucleated in a quantum fluid, they may be held in place using pinning potentials such as optical tweezers. The same laser beams may also be used for controllable steering of the vortices to braid their world lines. Fusion of two vortex anyons occurs when they are brought together in the same space–time event. Again, this can be achieved by overlapping the vortex-pinning laser beams. Physically, the outcome of the fusion is either annihilation of the two vortices or a formation of a new vortex particle which is a composite of the two vortices that were fused together.

9.11 From quantum circuits to anyon braiding FRS: anyon quantum circuits20 Conventional quantum computation may be coded using quantum circuit diagrams, such as shown in figure 9.2(a), that describe the operation of a certain quantum algorithm to be performed by a quantum computer. Such circuit diagrams are typically first decomposed and expressed in terms of a small set of elementary

Figure 9.2. (a) Quantum circuit Shor’s algorithm, (b) compilation to elementary gates, (c) mapping onto elementary braids, (d) concatenation of the elementary braids to yield topological implementation of the full quantum circuit.

20 B Field and T Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 3, 045004 (2018).

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unitary gate operations, such as those in figure 9.2(b). Depending on the physical architecture of the quantum computer, the realisation of the elementary gates may correspond to a variety of physical operations such as the application of a sequence of pulses of radio frequency electromagnetic radiation to manipulate the spin state of an electron or an atomic nucleus, or directing coherent laser light through lenses and beam splitters in optical systems. In a topological quantum computer, the physical action of an elementary quantum gate corresponds to physically moving (braiding) the anyons (vortices) around each other in the 2D space thus creating knots and links into their world lines. To specify the precise paths along which the anyons must be moved, that is, determining what specific braiding operations correspond to the action of certain unitary gates, involves a compilation step— finding a mapping between the unitary gates and their representative anyon braids; see figure 9.2(c). The final step is to concatenate the anyon braids of each elementary gate according to the circuit diagram, and in a specific order, to realize topological implementation of the complete quantum circuit. If error correction is desired, this can be achieved by adding any such error-correcting algorithms to the original quantum algorithm at the circuit level.

9.12 Evaluation of space–time knot invariants FRS: Quantum field theory and the Jones polynomial21, Aharonov-JonesLandau algorithm22, an exact algorithm23 Unraveling a knot is an exponentially hard problem since every crossing that is undone in the calculation of the Kauffman bracket polynomial doubles the number of required computational steps. This fact is at the core of many cryptographic algorithms and also underlies the principle of topological quantum information processing protocols: tying a knot is a polynomially simple operation requiring only p moves whereas unraveling an existing knot is exponentially hard requiring 2p number of moves. Identifying a knot by calculating its characteristic knot invariant such as the Jones polynomial is thus, for an arbitrarily complicated knot, a problem that requires a time that grows exponentially with the complexity of the knot, and therefore cannot be solved using any conventional digital computer. A topological quantum computer may be used for evaluating the Jones polynomial V (t ) of a knot at the complex roots t = e i 2π /k of unity, where k is integer, using the Aharonov–Jones–Landau algorithm that solves a BQP complete problem. In general, this involves compiling an elaborate braid or a weave that approximates a specific unitary matrix whose trace can be computed efficiently using 21 E Witten, Quantum field theory and the Jones polynomial, Communications in Mathematical Physics 121, 351 (1989). 22 D Aharonov, V Jones, and Z Landau, A Polynomial Quantum algorithm for Approximating the Jones Polynomial, Algorithmica 55, 395 (2009). 23 B Field and T Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 3, 045004 (2018).

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a Hadamard test in a quantum computer. However, with an anyon-model-dependent (e.g. k = 4 for Ising anyons, k = 5 for Fibonacci anyons) judicious choice of the point t, a simpler and exact algorithm can be implemented for which the actual space–time knot to be performed with the anyons is identical to the physical knot under consideration. Specifically, the magnitude of the Jones polynomial of any knot or a link defined by a plat closure of a braid, B pl , is n −1

∣VB pl(e i 2π k )∣ = d k2

Pr(∣0〉k ) ,

(9.18)

where dk = 2 cos(π /k ) is the quantum dimension of one of the non-abelian Model k anyons, n is the number of strands in the braid, and Pr(∣0〉k ) is the probability that the anyons fuse to vacuum after being braided. Due to its simplicity, the exact algorithm, equation (9.18), seems ideally suited for a proof of concept demonstration of the operation of a topological quantum computer.

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Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 10 Two-dimensional quantum turbulence

Superfluid turbulence means turbulence in superfluids, whose inherently quantum mechanical microscopic dynamics has an observable influence on the macroscopic properties of the turbulent flow. Broadly, such turbulence may be categorized in two distinct types: hydrodynamic turbulence characterized by vortices and their chaotic dynamics, and wave turbulence characterized by interference and nonlinear mixing of waves. Quantum turbulence refers specifically to the case where vortex degrees of freedom are crucial to the non-equilibrium dynamics of the superfluid.

10.1 Regular and chaotic few-vortex dynamics FRS: point vortex dynamics1,2,3 A system of N-point vortices in an unbounded domain has four first integrals (constants of motion). These conserved quantities are the energy N −1 N

ρ H=− s ∑ 2π i = 1

∑ sisj ln(∣ri − rj ∣),

(10.1)

j=i+1

two Cartesian components of linear impulse and an angular impulse:

1

H Aref, Point vortex dynamics: A classical mathematics playground, Journal of Mathematical Physics 48 065401 (2007). 2 P K Newton, Point vortex dynamics in the post-Aref era, Fluid Dynamics Research 46 031401 (2014). 3 A V Murray, A J Groszek, P Kuopanportti, and T Simula, Hamiltonian dynamics of two same-sign point vortices, Physical Review A 93 033649 (2016).

doi:10.1088/2053-2571/aafb9dch10

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ª Morgan & Claypool Publishers 2019

Quantised Vortices

N

Q = κ∑sj xj , j=1

N

P = κ∑sj yj

N

and

j=1

(

)

I = κ∑sj x j2 + yj2 .

(10.2)

j=1

According to Noether’s theorem, these are associated, respectively, with the translation symmetries of time (H ) and space (Q, P ) and rotation invariance (I) of the system. In addition to these first integrals, the enstrophy and the individual circulations of the vortices are also time-independent quantities. If the number of degrees of freedom (equal to the number of vortices in this case) is at most equal to the number of first integrals, the system is said to be integrable. If there are more degrees of freedom than constants of motion, the system is non-integrable and may exhibit chaotic dynamics. Consequently, four (and less) neutral vortex systems are integrable in an unbounded system. In the presence of a circular boundary, translation invariance is lost and only two (H , I ) first integrals remain. In this case, a single vortex, N = 1 in the system will travel along a circular periodic orbit and the dynamics are regular. For N = 2 the number of vortices equals the number of conserved quantities and the dynamics remains periodic. However, chaotic advection may occur already for two vortices in the sense that the dynamics of a virtual tracer particle placed in the flow field may be chaotic. Already the two-vortex system exhibits non-trivial topological structure in the phase space. For three or more vortices the number of independent degrees of freedom of the vortex system exceeds the number of conserved quantities, the system is no longer integrable and chaotic vortex dynamics is possible. As the number of vortices in the system further increases, the complexity of the dynamics rapidly increases and the system is best characterised by the tools of statistical mechanics and turbulence.

10.2 Inverse energy and direct enstrophy cascades FRS: inverse energy cascade4,5,6,7, enstrophy cascade8 In two-dimensional (2D) turbulence the approximate conservation of energy Etot = ∫ E (k )dk and enstrophy Ω tot = ∫ k 2E (k )dk is predicted to lead to a double cascade where the energy is transported from the energy injection scale across length scales toward ever larger wave lengths piling up energy at the system scale, while

4

L F Richardson, Weather Prediction by Numerical Process, Cambridge University Press (1922). R H Kraichnan, Inertial Ranges in Two-Dimensional Turbulence, Physics of Fluids 10 1417 (1967). 6 G K Batchelor, Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence, The Physics of Fluids 12, 233 (1969). 7 R H Kraichnan and D Montgomery, Two-dimensional turbulence, Reports on Progress in Physics 43 547 (1980). 8 M T Reeves, T P Billam, X Yu, and A S Bradley, Enstrophy Cascade in Decaying Two-Dimensional Quantum Turbulence, Physical Review Letters 119, 184502 (2017). 5

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vorticity is increasingly concentrating on smaller length scales. The inverse energy cascade is anticipated to result in the energy spectrum in the inertial range

E (k ) ∝ k − 5 3

(10.3)

with the same exponent as the usual Kolmogorov spectrum of 3D turbulence. Simultaneously, a direct enstrophy cascade is predicted to yield, up to logarithmic corrections, an energy spectrum

E (k ) ∝ k − 3

(10.4)

over a second inertial range at wavenumbers greater than the energy injection scale. The inverse energy cascade may be associated with the growth of increasingly large eddies and Onsager vortices in the system. The direct enstrophy cascade may be associated with filamentation of vortex patches.

10.3 Vortex near-field spectrum FRS: vortex core spectrum9 At length scales smaller than the vortex core size ξ, the condensate density n(r ) ∝ r 2 and the superfluid velocity vs(r ) ∝ 1/r such that the kinetic energy density within the vortex core is E (r )dr ∝ n(r )vs(r )2rdr ∝ rdr . The momentum space energy spectrum is therefore

Ecore(k > kξ ) ∝ k −3.

(10.5)

Note that the exponent of this power law is equal to that predicted by the direct enstrophy cascade and therefore care must be taken not to confuse the spectral feature due to an isolated vortex core with that of an enstrophy cascade. However, if the direct enstrophy cascade is present, it may smoothly transition to the vortex core spectrum at the vortex core scale without developing a spectral bottleneck.

10.4 Vortex far-field spectrum FRS: dilute vortex limit9 At length scales much greater than the vortex core size ξ, the condensate density n(r ) is approximately constant and the superfluid velocity vs(r ) ∝ 1/r such that the kinetic energy density outside the vortex core is E (r )dr ∝ n(r )vs(r )2rdr ∝ dr /r . The momentum space energy spectrum is therefore

Efarfield(k < kξ ) ∝ k −1.

(10.6)

9 A S Bradley and B P Anderson, Energy Spectra of Vortex Distributions in Two-Dimensional Quantum Turbulence, Physical Review X 2 041001 (2012).

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If the vortex gas is sufficiently dilute, the kinetic energy of the flow at length scales between the vortex core size and inter-vortex spacing is dominated by the far field of individual vortices and may result in this scaling form.

10.5 Vortex dipole spectrum FRS: vortex dipole field9 At length scales much greater than the inter-vortex distance D, the condensate density n(r ) is approximately constant and the superfluid velocity of a vortex– antivortex pair is that of a dipole field vs(r ) ∝ 1/r 2 such that the kinetic energy density is E (r )dr ∝ n(r )vs(r )2rdr ∝ dr /r 3. Thus in the momentum space this yields the energy spectrum Edipole(k < kD) ∝ k , (10.7) which vanishes in the k → 0 limit in contrast to the case of non-zero total circulation in which case the spectrum has an infrared divergence.

10.6 Kolmogorov–Obukhov spectrum FRS: turbulence energy spectrum10 In a fully developed turbulent state, a three-dimensional (3D) superfluid is expected to mimic classical fluid dynamics with a characteristic energy spectrum 5 (10.8) EKO(k ) ∝ k − 3 . In classical 2D systems, the inverse energy cascade is predicted to result in an energy spectrum with the same −5/3 exponent as the 3D case. This universal scaling law is ubiquitous and may be arrived at on dimensional grounds by assuming that the energy spectrum E (k ) ∝ η β k α is only a function of the energy dissipation rate ∂E η = − ∂t and the wavenumber k, where the energies are expressed per unit mass. Dimensional consistency, [E (k )] = [E ][L ] = ([E ]/[T ]) β /[L ]α , where [E ] = [L ]2 /[T ]2 , immediately yields β = 2/3 and α = −5/3.

10.7 Onsager vortex spectrum FRS: Onsager vortex spectrum11,12

10

U Frisch, Turbulence, Cambridge University Press (1995). G Gauthier, M T Reeves, X Yu, A S Bradley, M Baker, T A Bell, H Rubinsztein-Dunlop, M J Davis, T W Neely, Giant vortex clusters in a two-dimensional quantum fluid Science 364, 1264–7 (2019). 12 S P Johnstone, A J Groszek, P T Starkey, C J Billington, T P Simula and Kristian Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid Science 364, 1267–71 (2019). 11

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The inverse energy cascade is predicted to pile up energy at the scale of the order of the system size—a process which consequently must self-terminate the cascade process. The result is the formation of large-scale coherent Onsager vortex structures. Ultimately, in the absence of dissipative mechanisms, each Onsager vortex structure could undergo a condensation transition and be described as a multiply quantised vortex state. Prior to condensation, the Onsager vortices exist in a state of coherent vortex clusters. Assuming the mean condensate density n(r ) to be constant within the cluster and, according to the Feynman rule, the superfluid velocity to be equal to that of a rigidly rotating body vs(r ) ∝ r , the kinetic energy density inside an Onsager vortex cluster would be E (r )dr ∝ n(r )vs(r )2rdr ∝ r 3dr . Thus in the momentum space such energy spectrum scales as

E OV(k > k OV ) ∝ k −5.

(10.9)

Strictly, the condensate density is not constant within the vortex cluster and the vortex distribution within the cluster is typically not uniform, unlike in the case of a vortex lattice, which results in deviations to this idealised consideration. In the far field of an isolated Onsager vortex the spectrum

E OV(k < k OV ) ∝ k −1

(10.10)

reduces to the spectral shape of a single vortex. In the far field of an Onsager vortex dipole the spectrum

E OVdipole(k < k OV ) ∝ k

(10.11)

reduces to the spectral shape of a vortex–antivortex dipole, and a generic multipole velocity field vs(r ) ∝ r −α paired with uniform condensate density would yield a farfield spectrum:

E OVmultipole(k < k OV ) ∝ k 2α−3.

(10.12)

10.8 Spin turbulence spectrum FRS: spin turbulence spectrum13 In spinor Bose–Einstein condensates (BECs) spin-waves or magnons affect the turbulent superfluid dynamics. In a ferromagnetic spin-1 BEC the spin–spin interaction energy has been found to have a scaling of the form

Espin(k ) ∝ k −7 3.

13

(10.13)

K Fujimoto and M Tsubota, Counterflow instability and turbulence in a spin-1 spinor Bose–Einstein condensate, Physical Review A 85 033642 (2012).

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However, it may be challenging to observe such a scaling experimentally since the energy of the mass currents of the superfluid typically dominate the energetics in such systems.

10.9 Helmholtz decomposition FRS: kinetic energy decomposition9 A standard result of vector calculus states that an arbitrary vector field F = E + B may be expressed as a sum of two orthogonal components E and B such that the other one is curl-free:

∇×E=0

(10.14)

∇ · B = 0.

(10.15)

and the other is divergence-free:

The field E may be referred to as compressible, phonon-like, electric-field-like or a longitudinal component. The field B may be referred to as incompressible, vortexlike, magnetic-field-like or a transverse component. The gauge freedom also allows one to express the vector field

F = ∇ϕ + ∇ × v ,

(10.16)

in terms of a scalar potential ϕ and a velocity vector potential v. For a scalar BEC the Helmholtz construction allows the kinetic energy Ekin of the superfluid to be expressed as a sum of three components:

Ekin = E inc + Ecomp + Eqp where, by defining us =

n(r ) vs = u sinc + u scomp, ∇ · u sinc = 0, and ∇ × u scomp = 0,

m m ∣u sinc∣2 dA ; Ecomp = 2 2 ℏ2 ∇2 n(r) Eqp = dA 2m n(r)

E inc =

(10.17)

∫

∫ ∣u scomp∣2 dA;

and (10.18)

∫

are the incompressible, compressible and quantum kinetic energies, respectively.

10.10 Enstrophy conservation and non-conservation FRS: cascade of enstrophy14

14 M T Reeves, T P Billam, X Yu, and A S Bradley, Enstrophy Cascade in Decaying Two-Dimensional Quantum Turbulence, Physical Review Letters 119, 184502 (2017).

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In a scalar BEC, circulation is quantized and therefore the enstrophy is equal to the total number of vortices in the system. Naïvely, one might expect the enstrophy to be conserved then because the superfluid has zero kinematic viscosity, which in incompressible classical fluids guarantees the conservation of the circulation of every individual fluid element. However, in a compressible BEC, vortices and antivortices may readily annihilate, reducing the vortex number and thereby the enstrophy. The process also reduces the incompressible energy of the system by converting the incompressible kinetic energy of the vortices into the compressible energy of the phonons. Since in classical turbulence the double inverse energy, direct enstrophy cascades are attributed to the conservation of energy and enstrophy in the inertial range where the effective kinematic viscosity is negligible, it is not obvious that similar physics should be observed in quantum turbulence where it is not so clear to what extent and under what circumstances should an approximate enstrophy conservation be satisfied. In a BEC the annihilation of vortices leads to two competing effects in these systems, which are related to the transverse (nondissipative) and longitudinal (dissipative) force components of the vortex dynamics. The dissipative mechanisms reduce the incompressible kinetic energy per vortex in the system driving a direct energy cascade. The evaporative heating mechanism increases the incompressible kinetic energy per vortex in the system driving an inverse energy cascade. Which of these two processes dominates the dynamics depends on the details of the system.

10.11 Evaporative heating of vortices FRS: vortex evaporative heating15,16 The incompressible kinetic energy of an isolated vortex–antivortex pair is approximately

Edipole = −

ρs κ 2 ⎛ ξ ⎞ ln ⎜ ⎟ , ⎝d ⎠ 2π

(10.19)

which is large when the pair is overlapping, separated by a large distance d and becomes vanishingly small in the limit d → ξ effectively corresponding to the situation that the pair is separated only by the vortex core radius ξ. In a system that conserves the incompressible kinetic energy of the flow the energy associated with a particular vortex–antivortex pair is transferred to the global flow field as the pair approaches each other. When the pair annihilates, the number of vortices in the system decreases from Nv to Nv − 2, and therefore the approximately conserved total energy per vortex increases according to the sequence 15 T Simula, M J Davis, and K Helmerson, Emergence of Order from Turbulence in an Isolated Planar Superfluid, Physical Review Letters 113 165302 (2014). 16 S P Johnstone, A J Groszek, P T Starkey, C J Billington, T P Simula and Kristian Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid Science 364, 1267–71 (2019).

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Ekin Ekin Ekin < < … Nv Nv − 2 Nv − 4

(10.20)

This is evaporative heating of vortices: the number of vortices decreases at every enstrophy non-conserving annihilation event yet the energy per vortex increases due to the approximate conservation of energy. There are two possible outcomes of this process for a system that has initially equal numbers of vortices and antivortices. Either all vortices and antivortices annihilate or the energy per vortex grows large enough to support large-scale Onsager vortices which spatially separate causing the vortex annihilations to cease such that the system becomes trapped in a nonequilibrium state. In the presence of a sufficiently strong dissipative vortex–phonon coupling mechanism, such processes will compete with the vortex evaporative heating reducing the energy per vortex. Whether the ultimate fate of such a vortex system will always be a state with Nv = 0 is presently unknown.

10.12 Point vortex model of turbulence FRS: point vortex turbulence17,18 Point vortex models are well suited for studies of 2D quantum turbulence. Vortex number conserving dissipative effects may be phenomenologically included via a gradient force in the direction of the vortex motion that results in a ‘viscous’ drag on vortices. The vortex number non-conserving effects, vortex–antivortex annihilation in particular, that lead to vortex evaporative heating may be modeled algorithmically by irreversibly removing a vortex and an antivortex from the system if the dynamics brings them within an annihilation distance dann from each other. Various degrees of sophistication may be added to this simple rule and stochastic vortex– antivortex pair creation may easily be included in a similar fashion. Although such point vortex models of turbulence can readily be used for characterising evaporative heating and formation of Onsager vortices, the microscopic dynamics due to vortex– sound interaction is inherently absent in such models even if the vortex annihilation mechanism is included phenomenologically.

10.13 Non-abelian two-dimensional quantum turbulence FRS: non-abelian quantum turbulence19,20

17

L Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6, 279 (1949). T Simula, M J Davis, and K Helmerson, Emergence of Order from Turbulence in an Isolated Planar Superfluid, Physical Review Letters 113 165302 (2014). 19 T Mawson, G Ruben, and T Simula, Route to non-Abelian quantum turbulence in spinor Bose–Einstein condensates, Physical Review A 91, 063630 (2015). 20 M Kobayashi and M Ueda, Topologically protected pure helicity cascade in non-Abelian quantum turbulence, arXiv:1606.07190 (2016). 18

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Turbulence in a system whose topological excitations include non-abelian vortices is fundamentally different from that of quantum turbulence in a system characterized by a scalar order parameter with abelian vortices. In the former case the collision of two vortices may lead to multiple different dynamical outcomes. Conservation laws and dimensionality play significant roles in the dynamics of turbulence and 2D quantum turbulence has drastically different features in comparison to its 3D counterpart. In the former, inverse energy, direct enstrophy double cascade is typically associated with the approximate conservation law of enstrophy, while in 3D the absence of the enstrophy constraint allows for a direct energy cascade with the celebrated Kolmogorov–Obukhov spectral scaling law. In 3D it has been shown that if the collision dynamics of vortices is non-abelian, a conservation of helicity results in a cascade of helicity, which has not been observed to occur in abelian quantum turbulence. Similarly, in 2D non-abelian quantum turbulence rungihilation events are anticipated to have significant ramifications to the spectral flow of energy in such systems.

10.14 Superfluid Reynolds number FRS: superfluid Reynolds number21,22,23 The Reynolds number

Re =

UL ν

(10.21)

is a dimensionless quantity that characterises the state of the fluid flow. Here U = ∣vf − vL∣ is a relative speed between the fluid flow and an object of size L immersed in such a fluid, and ν is the kinematic viscosity. Vortices typically begin to form in a flow with Re ≈ 10 and fully developed turbulence sets in for flows with Re ≈ 10 4 − 105. A superfluid does not have kinematic viscosity. However, Onsager noted that the quantum of circulation κ = h /m could serve for a definition of a superfluid Reynolds number:

Res =

UL . κ

(10.22)

Reeves and coworkers suggested that a Strouhal number could be used for redefining a superfluid Reynolds number as

21 Edited By: Per Chr Hemmer, Helge Holden, and Signe Kjelstrup Ratkje, The Collected Works of Lars Onsager, World Scientific Series in 20th Century Physics 17, 1088 (1996). 22 M T Reeves, T P Billam, B P Anderson, and A S Bradley, Identifying a Superfluid Reynolds Number via Dynamical Similarity, Physical Review Letters 114, 155302 (2015). 23 M T Reeves, T P Billam, X Yu, and A S Bradley, Enstrophy Cascade in Decaying Two-Dimensional Quantum Turbulence, Physical Review Letters 119, 184502 (2017).

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UcL , κ

Res =

(10.23)

where Uc = ∣U − vc∣ is the relative speed between the superfluid flow past an obstacle of size L and a superfluid critical speed vc .

10.15 Eddy turnover time FRS: eddy turnover time24 The eddy turnover time

Te =

L vL

(10.24)

is the characteristic time for a vortex structure of size L that involves a characteristic speed vL to undergo a significant internal deformation due to the relative motion of its constituent vortices. For an Onsager vortex cluster with Nv vortices and a radius R the characteristic speed is of the order of Nv /R such that Te ≈ R2 /Nv . Using the Feynman criterion, 2Ω = κn v , shows that the eddy turnover time

Te ≈

2πR 4π 2 R 2 = ΩR κ Nv

(10.25)

approximately characterises the time it takes for an individual vortex in an Onsager vortex cluster to traverse the circumference of the cluster.

10.16 Anomalous hydrodynamics of vortices FRS: anomalous vortex hydrodynamics25,26,27 In a superfluid, the quantised vortices puncture the condensate order parameter and have a locally circulating velocity field around them with a quantised circulation. This microscopic superfluid velocity field may be coarse grained by averaging the microscopic velocity field over small patches of fluid, resulting in a macroscopic continuous velocity field. The hydrodynamics of such an averaged superfluid velocity field of the vortex fluid is anomalous. The microscopic structure of the quantised vortices leave an imprint in their macroscopic hydrodynamics which 24

U Frisch, Turbulence, Cambridge University Press (1995). G Baym and E Chandler, The hydrodynamics of rotating superfluids. I. Zero-temperature, nondissipative theory, Journal of Low Temperature Physics 50, 57 (1983). 26 P Wiegmann and A G Abanov, Anomalous Hydrodynamics of Two-Dimensional Vortex Fluids, Physical Review Letters 113, 034501 (2014). 27 X Yu and A S Bradley, Emergent Non-Eulerian Hydrodynamics of Quantum Vortices in Two Dimensions, Physical Review Letters 119, 185301 (2017). 25

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features anomalous stresses absent in Eulerian hydrodynamics. A well-known consequence of this is that in a rigidly rotating vortex lattice the velocity of the vortices cannot be equal to the coarse grained superfluid velocity.

10.17 Negative absolute temperature FRS: negative absolute temperature28,29,30 The original concept of negative absolute temperature is due to Onsager. The fundamental thermodynamic relation

dE = TdS − PdV

(10.26)

for a system of fixed volume yields a definition of the inverse temperature

1 ∂S = T ∂E

(10.27)

in terms of the internal energy E and the entropy S. A system in equilibrium whose entropy is a decreasing function of energy therefore has a negative absolute temperature according to this definition. As such, the existence of negative absolute temperature systems depends on how the entropy is defined. In particular, for the Gibbs and Boltzmann entropies the microstates are counted differently. The Gibbs entropy is always a monotonic function of energy, whereas for systems with nonmonotonic density of states the Boltzmann entropy yields absolute negative temperatures.

10.18 Negative absolute vortex temperature FRS: negative absolute temperatures of vortices28,31,32 The phase space {qi , pi } of a 2D point vortex system is equal to the 2D real space {xi , yi } and therefore if the fluid that the vortices are immersed in is confined to an area A, the phase space of the point vortices is also bounded. Consequently, the number of microstates (distinguishable vortex configurations) is finite and the Boltzmann entropy has a maximum at a finite point vortex energy. Increasing the energy of the vortex system beyond the maximum entropy state will result in 28

L Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6, 279 (1949). N F Ramsey, Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures, Physical Review 103, 20 (1956). 30 E Abraham and O Penrose, Physics of negative absolute temperatures, Physical Review E 95, 012125 (2017). 31 G Gauthier, M T Reeves, X Yu, A S Bradley, M Baker, T A Bell, H Rubinsztein-Dunlop, M J Davis, T W Neely, Giant vortex clusters in a two-dimensional quantum fluid Science 364, 1264–7 (2019). 32 S P Johnstone, A J Groszek, P T Starkey, C J Billington, T P Simula and Kristian Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid Science 364, 1267–71 (2019). 29

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vortex configurations characterised by a negative absolute vortex temperature. Physically, the maximum vortex entropy states correspond to vortex configurations where the vortices are reasonably uniformly distributed over the area of the fluid. If the same sign vortices are then moved closer to each other, forming clusters of same sign vortices, the mean energy of the state will increase but the Boltzmann entropy of the configuration will decrease. Thus such clustered equilibrium vortex configurations are characterised by a negative absolute temperature. Such negative vortex temperature states have been observed in experiments on 2D quantum turbulence in BECs.

10.19 Non-thermal fixed point FRS: anomalous non-thermal fixed point33,34 In a turbulent 2D superfluid the mean nearest-neighbour inter-vortex distance 〈d nn〉 has been predicted to evolve according to a self-similar scaling law. Upon approaching a thermal Gaussian fixed point a scaling law of the form

〈d nn〉 ∝ t1 2,

(10.28)

where t denotes time, has been predicted with an associated dynamical critical exponent z = 2. In contrast, upon evolving toward a non-thermal anomalous fixed point a scaling law of the form

〈d nn〉 ∝ t1 5

(10.29)

has been predicted to emerge, associated with a dynamical critical exponent z = 5. Depending on the initial conditions, the system may dynamically cross-over from one type of scaling to the other. Evidence for such a dynamical phase transition (cross-over in the case of small vortex numbers) has been obtained in experiments on 2D quantum turbulence in BECs.

10.20 Dynamical phase transitions FRS: dynamical quantum phase transitions35 A dynamical quantum phase transition is characterised by the Loschmidt amplitude

G(t ) = 〈ψ0∣e−iHt ℏ∣ψ0〉

(10.30)

becoming non-analytical at a critical time tc. Since the point vortex Hamiltonian M Karl and T Gasenzer, Strongly anomalous non-thermal fixed point in a quenched two-dimensional Bose gas, New Journal of Physics 19, 093014 (2017). 34 S P Johnstone, A J Groszek, P T Starkey, C J Billington, T P Simula and Kristian Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid Science 364, 1267–71 (2019). 35 M Heyl, Dynamical quantum phase transitions: a review, Reports on Progress in Physics 81 054001 (2018). 33

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Quantised Vortices

N −1 N

H=−

ρs ∑ 2π i = 1

∑ sisj ln(∣ri − rj ∣)

(10.31)

j=i+1

has a simple logarithmic dependence on the inter-vortex distances, the rate function

ln(∣G(t )∣2 ) N →∞ N

λ(t ) = − lim

(10.32)

becomes non-analytical if the dynamical exponent z that characterises the mean inter-vortex distance 〈d nn〉 ∝ t1/z changes in the course of the evolution of the system. Specifically, a dynamical phase transition is expected to occur if the dynamical critical scaling exponent changes from z = 5 (non-thermal anomalous fixed point) to z = 2 (Gaussian fixed point) at a critical time tc, as seems to be the case for a decaying 2D quantum turbulence.

10.21 Condensation of Onsager vortices FRS: vortex condensate36,37 The very high energy vortex states of a confined 2D system corresponds to vortex configurations where all vortices are found in clusters of same sign Onsager vortices. At the highest energies such Onsager vortex clusters may form vortex condensates when the vortex cores, specifically the kelvon quasiparticles associated with the vortices, begin to significantly overlap. A free-energy argument based on the energy cost of forming a giant vortex of multiple circulation quanta may be used for obtaining a critical temperature for the Onsager vortex condensation transition. Defining an effective one-dimensional harmonic oscillator frequency ωeff = ω KmK /m, where ω K is the kelvon frequency and mK is the kelvon-based vortex mass, the Bose– Einstein condensation of Onsager vortices has a critical temperature

1 kBTEBC = − ℏωeff N , 2

(10.33)

where N is the number of vortices in a system with all vortices of the same sign. The critical temperature may also be expressed in terms of the superfluid density ρs as

TEBC = −

ρs κ 2 N. 8πkB

(10.34)

36 T Simula, M J Davis, and K Helmerson, Emergence of Order from Turbulence in an Isolated Planar Superfluid, Physical Review Letters 113 165302 (2014). 37 R N Valani, A J Groszek, and T P Simula, Einstein-Bose condensation of Onsager vortices, New Journal of Physics 20, 053038 (2018).

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IOP Concise Physics

Quantised Vortices A handbook of topological excitations Tapio Simula

Chapter 11 Vortex states of matter in Flatland

Physical systems that may be described in terms of complex-valued order parameters are ubiquitous and all such systems may host vortices and antivortices. In many twodimensional (2D) systems the structure and dynamics of vortices becomes a crucial ingredient determining the physical properties and behaviours of such systems. Numerous vortex-hosting systems have been discovered, including exciton–polariton condensates, coherent electromagnetic fields, liquid crystals, magnetic materials, Bose–Einstein condensates (BECs), superfluid Fermi gases, helium superfluids and superconductors. In this chapter a small selection of such systems are mentioned.

11.1 BCS superconductivity FRS: Cooper pairs1, BCS theory2

In the Bardeen–Cooper–Schrieffer (BCS) picture of superconductivity, electrons form Cooper pairs whose mass, charge and spin are those of two electrons combined. Such a Cooper pair is a boson despite its constituent electrons being spin-1/2 fermions. Superconductivity arises due to Bose–Einstein condensation of Cooper pairs. The supercurrents arise as the fluid motion of such a superfluid composed of charged particles. The BCS superconductors in an external magnetic field may be modeled in terms of Bogoliubov–de Gennes (BdG) quasiparticles with a MBCS BdG matrix:

1

L N Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Physical Review 104 1189 (1956). J Bardeen, L N Cooper, and J R Schrieffer, Microscopic Theory of Superconductivity, Physical Review 106 162 (1957). 2

doi:10.1088/2053-2571/aafb9dch11

11-1

ª Morgan & Claypool Publishers 2019

Quantised Vortices

MBCS

2 ⎛ ℏ2 ⎛ ⎞ ie ⎞ ⎜− ⎟ A⎟ − EF + U (r) Δ(r) ⎜∇ − ℏc ⎠ ⎜ 2m ⎝ ⎟ =⎜ ⎟. (11.1) 2 ⎛ ⎞2 ie ℏ ⎜ A⎟ + EF − U *(r)⎟⎟ Δ*(r) ⎜∇ + ⎜ 2m ⎝ ℏc ⎠ ⎝ ⎠

The pair potential Δ = 〈ψψ 〉, the superconducting gap function, is related to the critical temperature TcBCS of superconductivity via TcBCS ≈ Δ/kB. There are no available quasiparticle states within ±Δ around the Fermi energy EF. The material superconducts because of the energy gap surrounding the Fermi sea such that there are no quasiparticle states which could be excited by the motion of the Cooper pairs until their kinetic energy exceeds the gap energy. The critical current is to a superconductor what the Landau critical velocity is to a superfluid. The sample stops superconducting if the value for the critical field or critical temperature is exceeded, causing all Cooper pairs to break up.

11.2 Meissner effect FRS: Meissner-Ochsenfeld effect3 Conventional superconductors in the presence of an external magnetic field feature the Meissner effect. The magnetic field drives dissipationless supercurrents on the surface of the material such that the induced magnetic field destructively interferes with the external magnetic field resulting in vanishing total magnetic field inside the superconducting material. As such, the superconductor repels external magnetic fields. Only when a sufficiently strong magnetic field value is exceeded will the magnetic field penetrate the superconductor by breaking the Cooper pairs into their constituent electrons. In type-I superconductors the whole sample loses its superconducting property when the critical magnetic field is exceeded, whereas in type-II superconductors the magnetic field first penetrates the superconductor along the cores of quantised vortices allowing the space in between the vortices to remain superconducting, and only when the density of vortices increases sufficiently in the sense that the vortex cores become overlapping will the superconductor turn into a normal state.

11.3 Type-II superconductors FRS: Type-II superconductivity4

3 W Meissner and R Ochsenfeld, Ein neuer Effekt bei Eintritt der Supraleitfähigkeit, Naturwissenschaften 21 787 (1933). 4 J N Rjabinin and L W Shubnikov, Magnetic Properties and Critical Currents of Supra-conducting Alloys, Nature 135 581 (1935).

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Quantised Vortices

Although the details of the microscopic mechanism responsible for high-temperature superconductivity remains unknown, their high critical temperature is thought to be underpinned by vortex dynamics in 2D systems. The circulation

−

c e

∮ p · d l = n Φ0 ,

(11.2)

where p = m v − e A/c is the canonical momentum and A is the vector potential, is quantised in units of the flux quantum

Φ0 =

hc , 2e

(11.3)

which is the counterpart of the circulation quantum κ in neutral superfluids. The factor of 2 is due to the supercurrent being carried by Cooper pairs consisting of two electrons.

11.4 Abrikosov vortex lattice FRS: vortex lattice5 In a uniform magnetic field, type-II superconductors are penetrated by quantised vortices each carrying one flux quantum of magnetic flux. The minimum energy configuration of such vortices is typically a (triangular) lattice, referred to as an Abrikosov lattice. Depending on the details of the pairing functions, lattice geometries other than triangular are possible.

11.5 Vortex pinning and creep motion FRS: vortex creep and pinning6 Superconducting materials are usually littered with defects, which act as pinning sites for the vortices. This allows the vortices to be pinned into static vortex lattice structures. In the absence of such pinning effect the Lorentz force due to the electromagnetic fields would cause the vortices to move. At sufficiently high temperature, thermal excitations are able to knock out the vortices from their pinning sites, resulting in creep motion of the vortices. This results in an effective resistance in the superconductor that eventually becomes strong enough that the superconducting property of the material disappears. Stronger pinning of the 5 A A Abrikosov, The magnetic properties of superconducting alloys, Journal of Physics and Chemistry of Solids 2 199 (1957). 6 G Blatter, M V Feigel’man, V B Geshkenbein, A I Larkin, and V M Vinokur, Vortices in high-temperature superconductors, Reviews of Modern Physics 66, 1125 (1994).

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vortices or a higher effective mass of vortices would enable the superconductors to withstand higher critical currents.

11.6 Vortex matter in rotating superfluids FRS: rotating bosons7, rotating fermions8, rotating neutron stars9,10 Simple rotating BECs at zero temperature may be modeled in terms of BdG quasiparticles with a MBose BdG matrix. In the frame rotating at the angular frequency Ω ⎛ 1 ⎞ (− i ℏ∇ − κ A) 2 − μ + U (r) + κ Φ(r) − Δ(r) ⎜ ⎟ 2 m ⎟, (11.4) MΩ = ⎜ 1 ⎜ ⎟ ⎜ (− i ℏ∇ − κ A) 2 + μ − U *(r) − κ Φ(r) ⎟ Δ*(r) − ⎝ ⎠ 2m

where the pair potential Δ(r) = gψ 2(r) is determined by the coupling constant g and the condensate wavefunction ψ (r). The vector and scalar potentials are

A(r) =

m Ω×r κ

and

Φ( r ) = −

κ ∣A(r)∣2 . 2m

(11.5)

The zero-energy BdG mode satisfies the Gross–Pitaevskii equation in the rotating frame:

⎛ ℏ2 ⎞ μψ (r) = ⎜ − ∇2 − ΩLz + U (r)⎟ψ (r) ⎝ 2m ⎠ ⎡ 1 ⎤ 1 ( −i ℏ∇ − κ A)2 − mΩ2r 2 + U (r)⎥ψ (r) =⎢ ⎦ ⎣ 2m 2

(11.6)

where Lz is the z-component of the orbital angular momentum operator. Hence there exists a duality between the electromagnetic field in a charged superfluid and an angular rotation frequency in a neutral superfluid. In charged superfluids, both rotation and magnetic fields result in the nucleation of flux tubes (vortices) in the superfluid. It is thought that the pulsar glitches in rotating neutron stars may be associated with quantum vortices likely present in the nuclear matter superfluids.

7 K W Madison, F Chevy, W Wohlleben, and J Dalibard, Vortex Formation in a Stirred Bose–Einstein Condensate, Reviews of Modern Physics 84, 806 (2000). 8 M W Zwierlein, J R Abo-Shaeer, A Schirotzek, C H Schunck, and W Ketterle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature 435, 1047 (2005). 9 G Baym, C Pethick and D Pines, Spin Up in Neutron Stars: The Future of the Vela Pulsar, Nature 224, 673 (1969). 10 P D Lasky, Gravitational Waves from Neutron Stars: A Review, Publications of the Astronomical Society of Australia 35, E034 (2015).

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11.7 Vortex nucleation and Hess–Fairbank effect FRS: Hess-Fairbank effect11,12 Rotation has a similar effect on a neutral superfluid as an external magnetic field has on a charged superconductor. In a rotating superfluid the Meissner effect is replaced by the Hess–Fairbank effect associated with the non-classical rotational inertia of the superfluid, such that for low external rotation frequencies the bulk superfluid remains stationary and only begins to rotate above a critical angular frequency Ωc . As with superconductors, the superfluid may be viewed to produce superflow, confined in the vicinity of the boundary of the superfluid, that decouples the bulk superfluid from the rotating drive. Once the critical rotation frequency Ωc = min(Eq /ℓq ), where Eq is the energy of a quasiparticle with orbital angular momentum ℓq, will the superfluid begin to rotate by nucleating quantised vortices. This picture holds true also at zero temperature and as such the vortex nucleation is inherently a quantum phase transition associated with a change in the topology of the superfluid order parameter. The transitions associated with the addition of each new vortex involves closing a gap between a surfon edge mode in the system and the condensate zero mode. A vortex may be nucleated from the periphery of the cloud when a surfon with a certain angular momentum becomes resonant with the condensate. A vortex–antivortex pair may be nucleated within the bulk of the superfluid by creating two ‘zero’ modes—one for each vortex.

11.8 Vortex lattices in neutral superfluids FRS: rotating vortex lattices13 Above the critical angular frequency, the orbital angular momentum of the superfluid increases with increasing external rotation frequency and ever more vortices are nucleated in the superfluid. In equilibrium at low temperatures the discretised vorticity field of the superfluid forms a crystalline lattice structure. This occurs for two energetic reasons: (i) the kinetic energy of a single vortex grows quadratically with its circulation winding number Ev ∝ ℓ 2 such that occurrence of vortices with a charge ℓ > ∣1∣ is improbable, (ii) the total kinetic energy of the superfluid has a minimum when the positions of the vortices form a crystal lattice because of the effective long-range interaction between vortices. The most typical vortex lattice structure is a triangular lattice, often referred to as an Abrikosov lattice even though Abrikosov’s original article predicted a square lattice as the minimum energy configuration for scalar 11

G B Hess and W M Fairbank, Measurements of Angular Momentum in Superfluid Helium, Physical Review Letters 19, 216 (1967). 12 A J Leggett, Quantum Liquids, Oxford University Press (2006). 13 A L Fetter, Rotating trapped Bose–Einstein condensates, Reviews of Modern Physics 81, 647 (2009).

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vortices that reflects the cylindrical symmetry of the individual vortices and therefore has the same structure as close packing of identical cylindrical rods. Other ground state lattice structures emerge in multicomponent systems and in the presence of external potentials and may be understood by considering the shape of the unit cell and mapping on to the close packing problem of identical objects.

11.9 Feynman rule FRS: Feynman rule for areal vortex density14 Bohr’s correspondence principle states that in the limit of large quantum numbers, quantum systems are expected to show classical behaviours. In this spirit, rotating neutral superfluids are able to mimic classical solid body rotation by nucleating a large number of quantum vortices which in equilibrium organise into a lattice structure. In a classical uniformly rotating system the vorticity ω = ∇ × v = 2Ω, where Ω is the orbital angular frequency of the rigidly rotating object. If a superfluid with a large number of vortices Nv is rotating at angular frequency Ω, we may expect on the basis of Bohr’s correspondence principle that

2ΩA =

∫

ωdA =

∫

Nv

κ∑δ(r − rv )dA = κNv.

(11.7)

i

This results in the Feynman rule for the mean areal vortex density:

nv =

Nv mΩ . = A πℏ

(11.8)

In practice, this relation works rather well not only for vortex lattices but also for non-equilibrium vortex distributions where the vortices are not spatially organised in a lattice structure as long as the vortex density remains reasonably uniform.

11.10 Vortex lattice melting FRS: lattice melting15,16 A vortex lattice, just as any other crystal structure, may undergo a melting transition when the energy contained in the excitations of the lattice is sufficiently high. A vortex lattice in a rotating BEC is also expected to melt at zero temperature if it rotates sufficiently rapidly. This is predicted to occur when the frequency of the Tkachenko modes of the vortex lattice become so small that quantum fluctuations 14

R P Feynman, Chapter II Application of Quantum Mechanics to Liquid Helium, Progress in Low Temperature Physics 1, 17 (1955). 15 S A Gifford and G Baym, Dislocation-mediated melting in superfluid vortex lattices, Physical Review A 78, 043607 (2008). 16 N R Cooper, Rapidly rotating atomic gases, Advances in Physics 57, 539 (2008).

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are sufficient to excite them. Calculations suggest that the vortex lattice would melt with the Lindeman criterion Na /Nv ≲ 10. In words, the vortex lattice is anticipated to melt when the number of vortices Nv is comparable to the number of atoms Na in the superfluid within the relevant area of the system. For even larger rotation frequencies, strongly correlated quantum-Hall-like phases may emerge in such systems.

11.11 Two-dimensional vortex Coulomb gas FRS: statistical hydrodynamics17, two-dimensional Coulomb gas18 The functional form of the Coulomb interaction potential Φ(r ) is determined by the Poisson equation:

∇2 Φ(∣r − ri ∣) = −2πδ(r − ri ).

(11.9)

In particular, Φ(r ) depends on the dimension of space such that

Φ 3 D (r ) ∝ 1 r ,

Φ(r )2D ∝ ln(r ),

and

Φ1D(r ) ∝ r.

(11.10)

Specifically, the dimensionless point vortex Hamiltonian N ⎛ ri − rj ⎞ H = −∑ΓΓ ⎟, i j ln ⎜ ⎝ rc ⎠ i