Modern Classification Theory of Superconducting Gap Nodes [1st ed.] 9789813342637, 9789813342644

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Springer Theses Recognizing Outstanding Ph.D. Research

Shuntaro Sumita

Modern Classification Theory of Superconducting Gap Nodes

Springer Theses Recognizing Outstanding Ph.D. Research

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Shuntaro Sumita

Modern Classification Theory of Superconducting Gap Nodes Doctoral Thesis accepted by Kyoto University, Kyoto, Japan

123

Author Dr. Shuntaro Sumita Condensed Matter Theory Laboratory RIKEN Cluster for Pioneering Research Saitama, Japan

Supervisor Prof. Youichi Yanase Department of Physics Kyoto University Kyoto, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-33-4263-7 ISBN 978-981-33-4264-4 (eBook) https://doi.org/10.1007/978-981-33-4264-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To my parents.

Supervisor’s Foreword

Research of superconductivity entered a major paradigm shift in the 1980s because of the discovery of unconventional superconductors that cannot be described in the standard Bardeen-Cooper-Schrieffer (BCS) theory. The discovery stimulated theoretical research on strongly correlated quantum many-body systems and created a new direction of experimental condensed matter physics. Whereas conventional superconductors assumed in the BCS theory show a finite excitation gap in energy spectrum, unconventional superconductors may host a gapless quasiparticle excitation (gapless Bogoliubov quasiparticle). To observe the quasiparticles in superconductors, various experimental methods to access low-energy excitations have been developed. Since the gap structure of Bogoliubov quasiparticles may suggest a novel mechanism of superconductivity and spontaneous symmetry breaking, its research has supported synergistic developments in theory and experiment in the field of superconductivity. As a theoretical basis, a group theory completed in a review article by Sigrist and Ueda [Rev. Mod. Phys. 63, 239 (1991)] has been adopted for several decades. In this theory, the order parameter of superconductivity is classified based on the crystallographic point group, and the superconducting gap structure is deduced from it. However, as I wrote “deduced”, the results obtained there are not exact. In fact, there are known examples of superconducting gap structures that cannot be described by the Sigrist-Ueda theory. Therefore, development in a theory that can accurately determine the quasiparticle’s gap structure in superconductors has been awaited. On the other hand, in the modern condensed matter physics, research on topological phases with a nontrivial structure in wavefunctions (topological insulators/superconductors) is explosively advanced. A theoretical method developed there provides a new way to study the quasiparticle excitations. In his doctoral dissertation, Shuntaro Sumita gave a rigorous and almost complete classification of the quasiparticle excitations in superconductors with the complementary use of symmetry and topology. His theory not only includes all the known quasiparticle excitations, but also uncovers novel Bogoliubov quasiparticles. He proposes that they can actually be realized in some heavy electronic superconductors, an vii

viii

Supervisor’s Foreword

important class of unconventional superconductors. The dissertation contains classification tables by which we can automatically understand quasiparticle’s gap structure when crystalline symmetry and superconducting symmetry are given. Thus, this thesis will be practically useful for interpreting experimental results in the future. Kyoto, Japan June 2020

Youichi Yanase

Parts of this thesis have been published in the following journal articles: 1. Shuntaro Sumita, Takuya Nomoto, Ken Shiozaki, and Youichi Yanase, Classification of topological crystalline superconducting nodes on high-symmetry lines: Point nodes, line nodes, and Bogoliubov Fermi surfaces, Physical Review B 99, 134513 (2019). 2. Shuntaro Sumita and Youichi Yanase, Unconventional superconducting gap structure protected by space group symmetry, Physical Review B 97, 134512 (2018). 3. Shingo Kobayashi, Shuntaro Sumita, Youichi Yanase, and Masatoshi Sato, Symmetry-protected line nodes and Majorana flat bands in nodal crystalline superconductors, Physical Review B 97, 180504(R) (2018). 4. Shuntaro Sumita, Takuya Nomoto, and Youichi Yanase, Multipole Superconductivity in Nonsymmorphic Sr2 IrO4 , Physical Review Letters 119, 027001 (2017).

ix

Acknowledgements

I would like to thank Youichi Yanase for his support during my master and doctor courses. I am deeply grateful to him for providing me many opportunities to discuss condensed matter physics, giving me plenty of advice on the direction of my study, and teaching me about the desired attitude to establish good researches. Without his guidance, I could not have done this work. I would like to thank Takuya Nomoto, Shingo Kobayashi, Masatoshi Sato, and Ken Shiozaki, for many fruitful and stimulating discussions through collaborations on the topics of gap classification theory. I also appreciate Manfred Sigrist in ETH Zürich and Mark H. Fischer in the University of Zürich, for fruitful discussions and supporting me during staying in Zürich. I would like to thank the faculty in Condensed Matter Theory Group at Kyoto University, Norio Kawakami, Ryusuke Ikeda, Robert Peters, and Masaki Tezuka, for their support in various academic matters as well as fruitful discussions. I also have greatly benefited from my colleagues in the Condensed Matter Theory Group for everyday discussions on physics and other topics. Especially, I am grateful to Akito Daido for a lot of stimulating discussions and drinking parties. I appreciate the financial support from the Japan Society for the Promotion of Science (JSPS) through Research Fellowship for Young Scientists (JPSJ KAKENHI Grant No. JP17J09908). Lastly, I am deeply grateful to my parents Toru and Emiko, whose encouragement and advice have been a great source of support in learning not only physics but also many other things.

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview of the Sigrist–Ueda Classification Method . . . . 1.1.1 Classification Scheme . . . . . . . . . . . . . . . . . . . . . 1.1.2 Inadequacies of the Sigrist–Ueda Method . . . . . . . 1.2 Recent Progress in Superconductors with Multi-degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Nonsymmorphic Multi-sublattice Superconductors . 1.2.2 Multi-orbital Superconductors . . . . . . . . . . . . . . . . 1.3 Organization of This Thesis . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Group-Theoretical Classification of Superconducting Gap . 2.3 Topological Classification of Superconducting Gap . . . . . 2.4 Example: Space Group P21 =m . . . . . . . . . . . . . . . . . . . . 2.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Group-Theoretical Gap Classification . . . . . . . . . . 2.4.3 Topological Gap Classification . . . . . . . . . . . . . . . Appendix 1: Wigner Criterion . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2: Orthogonality Test . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Superconducting Gap Classification on High-Symmetry Planes 3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Gap Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Application to 59 Space Groups . . . . . . . . . . . . . . . . 3.2 Topological Classification of Symmetry-Protected Line Nodes and Majorana Flat Bands . . . . . . . . . . . . . . . . . . . . .

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Contents

3.2.1 Line Node at a General Position . . . . . . . 3.2.2 Line Node on a High-Symmetry Plane . . 3.2.3 Possible Majorana Flat Bands . . . . . . . . . 3.3 Example: Sr2 IrO4 in
þ þ
State . . . . . . . . . 3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Classification of
þ þ
and
þ
þ on Magnetic Multipole . . . . . . . . . . . . . . 3.3.3 Superconducting Gap Classification . . . . 3.3.4 Numerical Calculation . . . . . . . . . . . . . . Appendix: Other Remarks for Sr2IrO4 . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Orders Based . . . . .

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54 55 55 61 63

4 Superconducting Gap Classification on High-Symmetry Lines 4.1 Group-Theoretical Classification . . . . . . . . . . . . . . . . . . . . 4.2 Topological Classification . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 UPt3 (Space Group: P63 =mmc) . . . . . . . . . . . . . . . . . . . . . 4.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Gap Classification on K-H Line . . . . . . . . . . . . . . . 4.3.3 Model and Normal Bloch State . . . . . . . . . . . . . . . 4.3.4 Effective Orbital Angular Momentum . . . . . . . . . . . 4.3.5 Gap Structures Depending on Bloch-State Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 SrPtAs (Space Group: P63 =mmc) . . . . . . . . . . . . . . . . . . . 4.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Classification on K-H Line . . . . . . . . . . . . . . . . . . . 4.5 CeCoIn5 (Space Group: P4=mmm) . . . . . . . . . . . . . . . . . . 4.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Classification on C-Z Line . . . . . . . . . . . . . . . . . . . 4.6 UCoGe (Space Group: Pnma) . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Classification on C-Z Line . . . . . . . . . . . . . . . . . . . 4.7 MoS2 (Space Group: P63 =mmc) . . . . . . . . . . . . . . . . . . . . 4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Classification on K-H Line . . . . . . . . . . . . . . . . . . . 4.8 UBe13 (Space Group: Fm3c) . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Classification on C-L Line . . . . . . . . . . . . . . . . . . . 4.9 PrOs4 Sb12 (Space Group: Im3) . . . . . . . . . . . . . . . . . . . . . 4.9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Classification on C-P and P-H Lines . . . . . . . . . . . Appendix: Effective Orbital Angular Momentum Due to Sublattice Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Acronyms

0D 1D 2D 3D AFM ASOC AZ BCS BdG BP BZ CS dHvA FFLO FM FS IR IS MO MQ N/A NMR NQR PHS PM SOC TRS ZF lSR

Zero-dimensional One-dimensional Two-dimensional Three-dimensional Antiferromagnetic Antisymmetric spin-orbit coupling Altland-Zirnbauer Bardeen-Cooper-Schrieffer Bogoliubov-de Gennes Basal plane Brillouin zone Chiral symmetry de Haas van Alphen Fulde-Ferrell-Larkin-Ovchinnikov Ferromagnetic Fermi surface Irreducible representation Inversion symmetry Magnetic octupole Magnetic quadrupole Not applicable Nuclear magnetic resonance Nuclear quadrupole resonance Particle-hole symmetry Paramagnetic Spin-orbit coupling Time-reversal symmetry Zone face Muon spin rotation/relaxation

xvii

Chapter 1

Introduction

In most (conventional) superconductors, superconductivity is well described within the BCS theory in 1957 [1], which gives the microscopic picture of superconducting state as a condensation of bounded pairs of two electrons (Cooper pairs) on the FS. The BCS theory, in a narrow sense, has three assumptions about Cooper pairs: (i) zero center-of-mass momentum, (ii) zero total spin (S = 0; spin-singlet), and (iii) isotropic s-wave symmetry. Then, the resulting superconducting order parameter is represented as (k) = 0 iσ y , where σi is the Pauli matrix in a spin space. Here, 0 is a k-independent constant, which means that the order parameter opens a gap over the whole FS. On the other hand, recent studies have suggested unconventional superconductivity, which do not satisfy all the assumptions (i)–(iii). If the assumption (i) is broken, for example, the resulting state is called FFLO superconductivity [2–4], where Cooper pairs have a finite center-of-mass momentum. Other examples are (spinsinglet) anisotropic superconductivity without the assumption (iii), and spin-triplet superconductivity without (ii) and (iii) [5]. In anisotropic and spin-triplet superconductors, the order parameter has a momentum dependence (e.g., p-wave and d-wave symmetry), which may indicates the presence of gapless points in the superconducting state, called superconducting nodes. Such momentum dependence of the superconducting gap is closely related to the symmetry of superconductivity and the pairing mechanism. Since the superconducting gap structure can be identified by various experiments [6, 7], combined studies of the superconducting gap by theory and experiment may clarify the characteristics of superconductivity. In this context, classification of a superconducting gap is one of the central subjects in theoretical researches of unconventional superconductivity. Most of the previous studies have been based on the classification of an order parameter by the crystal point group [8–10], which was summarized by Sigrist and Ueda (called the Sigrist–Ueda method in this thesis) [11]. However, their classification may not provide a precise result of the superconducting gap, namely an excitation energy in the Bogoliubov quasiparticle spectrum. Indeed, recent studies have discovered unusual gap structures incompatible with the Sigrist–Ueda classification [12–32]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Sumita, Modern Classification Theory of Superconducting Gap Nodes, Springer Theses, https://doi.org/10.1007/978-981-33-4264-4_1

1

2

1 Introduction

In the following, we briefly explain the Sigrist–Ueda method, and reveal inadequacies of the classification scheme (Sect. 1.1). Then we concretely show the examples of unconventional superconducting gap structures elucidated in the recent studies (Sect. 1.2). Finally the organization of this thesis is represented in Sect. 1.3.

1.1 Overview of the Sigrist–Ueda Classification Method In this section, we overview the Sigrist–Ueda classification scheme of superconducting order parameters [8–11].

1.1.1 Classification Scheme In the discussion of the section, we assume the existence of spatial IS and TRS in the normal state. Thus, the normal-state band structure has a twofold (Kramers) degeneracy for any momentum k. Let us consider a BCS mean-field Hamiltonian, HBCS =



ξ(k)as† (k)as (k) +

1  [s1 s2 (k)as†1 (k)as†2 (−k) + H.c.], 2

(1.1)

k,s1 ,s2

k,s

where ξ(k) is the band energy measured relative to the chemical potential, and (k) is the superconducting order parameter. Since this Hamiltonian includes no SOC term, the Hamiltonian has a certain symmetry represented by a group P × SU(2), where P is the point group of the crystal lattice symmetry and SU(2) is the spin-rotation symmetry group. The behavior of (k) under a symmetry operation in P × SU(2) is determined by the invariance condition of HBCS . A point group operation p ∈ P acts only on the momentum k, pas† (k) p −1 = as† ( pk), ⇒ p : s1 s2 (k) → s1 s2 ( p

(1.2) −1

k).

(1.3)

On the other hand, the spin-rotation symmetry g ∈ SU(2) transforms the matrix space of (k) as gas† (k)g −1 =



as† (k)[D(1/2) (g)]s  s ,

(1.4)

s

⇒ g : s1 s2 (k) → [D(1/2) (g)(k)D(1/2) (g)T ]s1 s2 ,

(1.5)

where the 2 × 2 matrix D(1/2) is the representation of SU(2) in the pure spin-1/2 space. In the presence of SOC, however, the point-group transformation and the spin

1.1 Overview of the Sigrist–Ueda Classification Method

3

transformation can no longer be treated independently; the spin-rotation group SU(2) is absorbed by the point group P. Although the Bloch states cannot be eigenstates of the pure-spin operator anymore, they can be labeled by pseudospin bases; we dub the new Bloch basis as cσ† (k). Thus, the transformation property of the order parameter is changed as pcσ† (k) p −1 =

 σ

cσ†  ( pk)[D(1/2) (g)]σ  σ ,

⇒ p : σ1 σ2 (k) → [D(1/2) ( p)( p −1 k)D(1/2) ( p)T ]σ1 σ2 ,

(1.6) (1.7)

where p ∈ P is a point-group operation.1 According to Landau theory for the second-order phase transition, on one side of the transition point the crystal has the higher symmetry, while on the other side the symmetry is lower; the order parameter of the symmetry broken phase has to belong to an IR of the higher symmetry group in the other phase [33, 34]. In anisotropic superconductors, the point group symmetry P as well as the U(1) symmetry is broken below the transition temperature Tc , and thus, the order parameter belongs to an IR of P. Conversely speaking, superconducting order parameters can be classified by IRs of the point group P. In the following, we concretely see the classification method of the order parameter using the point group. First, let us consider the limit where the SOC and the crystalline electric field is negligibly small. In this case, the “point group” is represented by the rotation group: P = SO(3) × I with the IS I. Therefore, the superconducting order parameter is represented in the function space D(l) × D(S) , where D(l) denotes the k-space representation and D(S) the spin-space representation of P.2 As a concrete representation, the order parameter is given by the spherical harmonics Ylm (k) for D(l) , and the spin functions χs (0, 0) = 1 for D(S=0) and ⎧ ⎪ xˆ + i yˆ sz = −1 ⎨√ χt (1, sz ) = 2ˆz sz = 0 ⎪ ⎩ −xˆ + i yˆ sz = +1

(1.8)

for D(S=1) space. That is, ⎧ ˆ y ⎪ cm Ylm ( k)iσ singlet; l = even, ⎪ ⎨ m l (k) = ˆ ⎪ ˆ y triplet; l = odd, cm nˆ Ylm ( k)(σ · n)iσ ⎪ ⎩

(1.9)

ˆ x, m,n= ˆ yˆ ,ˆz

the spatial inversion I ∈ P is not an element of SU(2), we define D(1/2) (I ) as the identity matrix σ0 . 2 The indices l = 0, 1, 2, . . . and S = 0, 1 are relative angular momentum and total spin angular momentum, respectively. 1 Since

4

1 Introduction

Table 1.1 Classification of superconducting order parameters by even-parity basis functions i (k) = ψi (k)iσ y and odd-parity basis functions i (k) = d i (k) · σ iσ y for the tetragonal D4h group [11] IR (even) Basis functions IR (odd) Basis functions ψ(k) = 1, k x2 + k 2y , k z2 ψ(k) = k x k y (k x2 − k 2y ) ψ(k) = k x2 − k 2y ψ(k) = k x k y ψ1 (k) = k x k z ψ2 (k) = k y k z

A1g A2g B1g B2g Eg

A1u A2u B1u B2u Eu

d(k) = k x xˆ + k y yˆ , k z zˆ d(k) = k y xˆ − k x yˆ d(k) = k x xˆ − k y yˆ d(k) = k y xˆ + k x yˆ d 1 (k) = k z x, ˆ k x zˆ d 2 (k) = k z yˆ , k y zˆ

where cm and cm nˆ are complex numbers. If we introduce SOC, these function spaces split into subspaces due to the symmetry lowering. In order to obtain the new classification spaces, we have to decompose the Kronecker product of k- and spin-space, 

D(l) × D(S=0) = D(l) singlet; l = even, D(l) × D(S=1) = D(l−1) + D(l) + D(l+1) triplet; l = odd.

(1.10)

The basis functions of the new representations can be derived by the Clebsch–Gordan formalism, which can be labeled by the total angular momentum J . Now we turn the crystalline electric field on. The crystal symmetry is lowered from a continuous rotation group SO(3) × I with an infinite number of IRs to a discrete point group with only a few IRs. Thus, we can classify the basis functions of the order parameters, by considering projections of the functions belonging to D(J ) (with small J ) on the IRs  of the crystal point group P. For example, Table 1.1 shows the classification results for P = D4h summarized in Ref. [11]. From the table, the dx 2 −y 2 -wave order parameter is found to belong to the B1g IR of D4h , which guarantees the presence of line nodes on the symmetry planes k x ± k y = 0. Such a gap structure is proposed in cuprates [5] and CeCoIn5 [5, 35, 36].

1.1.2 Inadequacies of the Sigrist–Ueda Method In the previous section, we see the classification scheme of the Sigrist–Ueda method, which is useful for symmetry analysis of superconducting order parameters. Indeed, the classification tables for the basic point groups Oh , D6h , and D4h in Ref. [11] have been used to unveil the fundamental properties of various unconventional superconductivity. However, recent studies have elucidated that the results of the Sigrist–Ueda method may not be precise for actual superconducting gap structures [12–32]. In the following, we refer to three related problems of the method.

1.1 Overview of the Sigrist–Ueda Classification Method Fig. 1.1 Schematic picture of two-band (four-band in the particle-hole space) superconductivity. Each figure shows the Bogoliubov quasiparticle spectrum for a finite intra-band and zero inter-band order parameters, and b zero intra-band and finite inter-band order parameters

1.1.2.1

(a)

5

(b)

(Intra-band OP) ≠ 0

(Intra-band OP) = 0

(Inter-band OP) = 0

(Inter-band OP) ≠ 0

Incompatibility Between Order Parameter and Gap Structure

First problem is that the order parameter obtained by the method does not appropriately indicate the superconducting gap structure. For example, let us consider twoband superconductivity depicted in Fig. 1.1. Figure 1.1a shows the situation where the intra-band order parameter is finite, while the inter-band order parameter is zero. In this case, the amplitude of the order parameter corresponds to the size of the superconducting gap (an excitation energy in the Bogoliubov spectrum). In Fig. 1.1b, on the other hand, the situation is inverted; the intra-band (inter-band) order parameter is zero (finite). If the band splitting is sufficiently larger than the inter-band order parameter, the superconducting gap is zero in spite of the finite order parameter. This is a simple example of the incompatibility between an order parameter and a gap structure. From the above reason, our new classification method focuses only on intra-band order parameters (Chap. 2).

1.1.2.2

Absence of Space Group Symmetry Analysis

Secondly, the space group symmetry is not taken into account in the method. Since a space group is given by the combination of a point group and a translation group, it provides us with more information than the point group. Now we point out the importance of considering nonsymmorphic space group symmetry, by showing an example. In 1985, Blount showed that no symmetry-protected line node exists in oddparity superconductors [37], which is consistent with the result of the Sigrist–Ueda method. After that, however, some studies showed a counter-example, namely, a line node in nonsymmorphic systems, and indeed suggested a line node protected by a nonsymmorphic space group symmetry in UPt3 [13, 18–22]. At present, it is known that Blount’s theorem holds only in symmorphic crystals. The essence is that the nonsymmorphic symmetry causes the difference in the group-theoretical representation of gap functions between the BPs and the ZFs in the BZ. Although the Sigrist–Ueda method, which takes only point group symmetry into account,

6

1 Introduction

appropriately implies the gap functions on the BPs, it may fail to show those on the ZFs.

1.1.2.3

Absence of Higher-Spin State Analysis

Third issue is that the Sigrist–Ueda method considers only spin-1/2 states. In multiorbital and multi-sublattice superconductors, however, a band basis often acquires total angular momentum larger than 1/2. Such higher-pseudospin (3/2, 5/2, . . . ) states may affect the resulting superconducting gap structures. Indeed, recent theoretical studies have suggested novel type of superconducting order parameters in higher-spin states [25–32].

1.2 Recent Progress in Superconductors with Multi-degrees of Freedom In this section, we introduce recent developments beyond the results of the Sigrist– Ueda classification in the previous section. The unconventional results basically emerge in superconductors with multi-degrees of freedom, such as multi-sublattice (Sect. 1.2.1) and multi-orbital superconductors (Sect. 1.2.2).

1.2.1 Nonsymmorphic Multi-sublattice Superconductors As mentioned in Sect. 1.1.2, recent theoretical studies have pointed out that the oddparity superconductor UPt3 possesses line nodes characterized by nonsymmorphic symmetry [13, 18–22], which was not predicted by the Sigrist–Ueda method. After that, nonsymmorphic-symmetry-protected nodes have been suggested in other superconductors: UCoGe [23], URhGe [23], UPd2 Al3 [23, 24, 38], UNi2 Al3 [24], and CrAs [24]. In these superconductors, nonsymmorphic symmetry (namely screw or glide symmetry) ensures that the crystal structure includes more than one sublattice. Such sublattice degree of freedom often play essential roles on the nodal structure. In order to see the importance of sublattice degree of freedom, we consider an example of a 1D zigzag chain with two sublattice a and b, depicted in Fig. 1.2. The system has nonsymmorphic screw symmetry {C2z | 2zˆ },3 which transfers each a atom to the b atom at the neighboring unit cell (the green arrow in Fig. 1.2). The transformation of g = {C2z | 2zˆ } is represented as follows:

= { pg |t g } is a conventional Seitz space group symbol with a point-group operation pg and a translation t g .

3 The notation g

1.2 Recent Progress in Superconductors with Multi-degrees of Freedom

7

a

z b Fig. 1.2 A 1D zigzag chain with two sublattice a and b. The black rectangle and the red point represents the unit cell and the inversion center, respectively. Due to screw symmetry {C2z | 2zˆ }, each a atom is transferred to the b atom at the neighboring unit cell (green arrow) † gcaσ (k z )g −1 = e−ikz /2

=

 mσ 



† cbσ  (C 2z k z )[D(1/2) (g)]σ  σ

σ

cm†  σ  (k z )[D(perm) (g; k z )]m  a [D(1/2) (g)]σ  σ .

(1.11)

† (k z ) (m = a, b) is a creation operator for sublattice m and pseudospin where cmσ σ . Comparing Eq. (1.11) with Eq. (1.6), there newly appears [D(perm) (g; k z )]m  m = e−ikz /2 δm  ,gm , which is a representation matrix describing the interchange of the two sites.4 The momentum (k z ) dependence of the matrix, which is a consequence of nonsymmorphic symmetry, has a crucial role in the classification of superconducting gap functions. However, few works have investigated gap structures for general nonsymmorphic symmetries as well as some representative superconductors [22, 24].

1.2.2 Multi-orbital Superconductors The Sigrist–Ueda method performs classification of the superconducting order parameter based on the band-based representation, where the normal Bloch states are assumed to possess spin-1/2 degrees of freedom. Although the classification scheme is always possible, the electrons have other internal degrees of freedom such as orbital and sublattice, which cannot be neglected in real superconductors. Indeed, we have shown that sublattice degree of freedom affects the gap structure in the previous subsection. Now let us consider the effect of orbital degree of freedom on the transformation property of gap functions. In multi-orbital superconductors, the normal Bloch states have a local orbital as well as (pseudo)spin-1/2 degrees of freedom. Here, we suppose a real atomic orbital with angular momentum L. Then, the normal Bloch basis is represented by (L)† (k) with the orbital (l) and the pseudospin (σ ) bases, which the creation operator clσ † that cmσ (k z ) in Eq. (1.11) is defined by the sublattice-dependent Fourier transformation, while the sublattice-independent one is used in Appendix of Chap. 4.

4 Note

8

1 Introduction

transforms as (L)† (k) p −1 = pclσ

 lσ 

cl(L)†  σ  ( pk)[D(L) ( p)]l  l [D(1/2) ( p)]σ  σ ,

(1.12)

for a point-group operation p ∈ P. Here, D(L) ( p) is a representation matrix describing the transformation among the orbital bases with angular momentum L, which makes a difference from Eq. (1.6). The product of D(L) × D(1/2) generates the Bloch states with higher total angular momentum than 1/2, which play crucial roles on the superconducting gap structures. Indeed, the recent classification theory of order parameter in multi-orbital superconductors [25, 39] has suggested the nontrivial k dependence of the gap function. For example, multi-orbital even-parity superconductors can possess spin-triplet (and orbital-triplet) order parameter, which is forbidden in single-orbital even-parity superconductors. Furthermore, recent theoretical studies have investigated the pairing of Bloch states with j = 3/2 total angular momentum [26–32]. The theories have shown that j = 3/2 fermions permit Cooper pairs with quintet or septet total angular momentum, in addition to the usual singlet and triplet states. Also, a new type of nodal structure is founded to appear in the j = 3/2 system; in TRS breaking even-parity superconductors, the conventional nodal structure (a point or a line node) is “inflated,” which means the appearance of a surface node [27, 28]. They called it a Bogoliubov FS and elucidated that the Bogoliubov FS is characterized by a 0D Z2 topological number.

1.3 Organization of This Thesis In this thesis, we offer a comprehensive classification theory of superconducting gap nodes beyond the Sigrist–Ueda method [11] in various superconductors with SOC and multi-degrees of freedom. In Chap. 2, the modern classification method of superconducting gap, which makes use of the combination of group theory and topology, is introduced. Using the method, we systematically classify gap structures on high-symmetry regions in the BZ: mirror-invariant planes (Chap. 3) and rotation-invariant axes (Chap. 4). In Chap. 3, general conditions for nontrivial nonsymmorphic-symmetry-protected gap structures are shown from both aspects of finite-group representation theory [40] and topology [41]. As an example, such unusual gap structures are demonstrated in the model of Sr2 IrO4 [42]. In Chap. 4, on the other hand, the classification results on rotational axes show that unconventional nodal structures appear even in symmorphic superconductors. We emphasize that higher-spin 3/2 states affect the superconducting gap structures; the gap classification depends on the total angular momentum jz = 1/2, 3/2, . . . of normal Bloch states on threefold and sixfold rotational-symmetric lines [40, 43]. Such jz -dependent gap structures are also discussed in various candidate superconductors. Especially, we elucidate the presence of perfectly spin-polarized nonunitary superconductivity in UPt3 [43].

References

9

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

Bardeen J, Cooper LN, Schrieffer JR (1957) Phys Rev 108:1175 Fulde P, Ferrell RA (1964) Phys Rev 135:A550 Larkin AI, Ovchinnikov YN (1964) Zh Eksp Teor Fiz 47:1136, [Sov Phys JETP 20:762 (1965)] Matsuda Y, Shimahara H (2007) J Phys Soc Jpn 76(5):051005 Yanase Y (2003) Phys Rep 387(1):1 Matsuda Y, Izawa K, Vekhter I (2006) J Phys: Conden Matter 18(44):R705 Sakakibara T, Kittaka S, Machida K (2016) Rep Prog Phys 79(9):094002 Volovik GE, Gor’kov LP (1984) Pis’ma Zh Eksp Teor Fiz 39:550 Volovik GE, Gor’kov LP, Eksp Zh (1985) Teor Fiz 88:1412 Anderson PW (1984) Phys Rev B 30:4000 Sigrist M, Ueda K (1991) Rev Mod Phys 63:239 Yarzhemsky VG, Murav’ev EN (1992) J Phys: Condens Matter 4(13):3525 Norman MR (1995) Phys Rev B 52:15093 Yarzhemsky VG (1998) Phys Status Solidi B 209(1):101 Yarzhemsky VG (2000) Int J Quant Chem 80(2):133 Yarzhemsky VG (2003) Conf AIP Proc 678(1):343 Yarzhemsky VG (2008) J Opt Adv Mater 10(7):1759 Micklitz T, Norman MR (2009) Phys Rev B 80:100506(R) Kobayashi S, Yanase Y, Sato M (2016) Phys Rev B 94:134512 Yanase Y (2016) Phys Rev B 94:174502 Nomoto T, Ikeda H (2016) Phys Rev Lett 117:217002 Micklitz T, Norman MR (2017) Phys Rev B 95:024508 Nomoto T, Ikeda H (2017) J Phys Soc Jpn 86(2):023703 Micklitz T, Norman MR (2017) Phys Rev Lett 118:207001 Nomoto T, Hattori K, Ikeda H (2016) Phys Rev B 94:174513 Brydon PMR, Wang L, Weinert M, Agterberg DF (2016) Phys Rev Lett 116:177001 Agterberg DF, Brydon PMR, Timm C (2017) Phys Rev Lett 118:127001 Timm C, Schnyder AP, Agterberg DF, Brydon PMR (2017) Phys Rev B 96:094526 Savary L, Ruhman J, Venderbos JWF, Fu L, Lee PA (2017) Phys Rev B 96:214514 Boettcher I, Herbut IF (2018) Phys Rev Lett 120:057002 Kim H, Wang K, Nakajima Y, Hu R, Ziemak S, Syers P, Wang L, Hodovanets H, Denlinger JD, Brydon PMR, Agterberg DF, Tanatar MA, Prozorov R, Paglione J, Sci Adv Venderbos JWF, Savary L, Ruhman J, Lee PA, Fu L (2018) Phys Rev X 8:011029 Landau LD, Lifshitz EM (2013) Statistical physics part 1. Course of theoretical physics, vol 5, 3rd edn. Elsevier, Oxford Inui T, Tanabe Y, Onodera Y (1990) Group theory and its applications in physics. Springer series in solid-state sciences, vol 78. Springer, Berlin Izawa K, Yamaguchi H, Matsuda Y, Shishido H, Settai R, Onuki Y (2001) Phys Rev Lett 87:057002 Izawa K, Yamaguchi H, Matsuda Y, Sasaki T, Settai R, Onuki Y (2002) J Phys Chem Solids 63(6):1055 Blount EI (1985) Phys Rev B 32:2935 Fujimoto S (2006) J Phys Soc Jpn 75(8):083704 Wan Y, Wang QH (2009) Europhys Lett 85(5):57007 Sumita S, Yanase Y (2018) Phys Rev B 97:134512 Kobayashi S, Sumita S, Yanase Y, Sato M (2018) Phys Rev B 97:180504(R) Sumita S, Nomoto T, Yanase Y (2017) Phys Rev Lett 119:027001 Sumita S, Nomoto T, Shiozaki K, Yanase Y (2019) Phys Rev B 99:134513

Chapter 2

Method

2.1 Preparation In this section, we define some terminologies and notations which are commonly used in the group-theoretical classification (Sect. 2.2) and the topological classification (Sect. 2.3). In all the discussions below, we assume centrosymmetric superconductors,1 which have the IS I = {I |0}. Also, PM and AFM superconductors have the TRS T = {T |t T }, while FM ones possess no TRS. Note that t T represents a non-primitive translation for AFM superconductors, while it is zero for PM superconductors. First, let M a magnetic space group of a system; suppose that G is a unitary space group including IS, M is equal to G + T G for PM and AFM, and is G for FM superconductors. M and G include a translation group T, which defines a Bravais lattice. In order to classify the gap structure on high-symmetry k points (mirror planes and rotation axes) in the BZ, we focus on the operations in G (M) leaving the k points invariant modulo a reciprocal lattice vector, which generates a subgroup of G (M). The subgroup G k < G (Mk < M) is called a (magnetic) little group. The factor group of the (magnetic) little group by the translation group G¯ k = G k /T ¯ k = Mk /T) is called a (magnetic) little cogroup. The (magnetic) little cogroup (M ¯ k = Cs + is isomorphic to the corresponding (magnetic) point group; G¯ k = Cs (M k k ¯ = Cn + T ICn or Cnv + T ICnv ) T ICs ) for mirror planes, and G¯ = Cn or Cnv (M for n-fold rotational axes (n = 2, 3, 4, and 6). Here, we define λαk as a double-valued small corepresentation of the magnetic little group Mk , which represents the normal Bloch state with the crystal momentum k: † mcαi (k)m −1 =



cα† j (k)[λαk (m)] ji ,

(2.1)

j

1 Indeed,

our topological classification theory (Sect. 2.3) can also be applied to other cases, e.g., noncentrosymmetric superconductors. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Sumita, Modern Classification Theory of Superconducting Gap Nodes, Springer Theses, https://doi.org/10.1007/978-981-33-4264-4_2

11

12

2 Method

where m ∈ Mk . α is a label of the double-valued IR, which corresponds to the total angular momentum of the Bloch state jz = ±1/2, ±3/2, . . . in spin-orbit coupled systems.2 Since Mk is a semi-direct product between the magnetic little cogroup ¯ k and the translation group T, the small corepresentation satisfies the following M equation for the translation subgroup T < Mk : λαk (TR ) = e−i k·R ,

(2.2)

where TR = {E|R} ∈ T. Furthermore, λαk has a factor system {ωin (l1 , l2 )} ∈ H 2 (M, U(1)φ ), which arises from internal degrees of freedom (e.g. a half-integer spin of electrons):  ωin (m 1 , m 2 )λαk (m 1 m 2 )

=

λαk (m 1 )λαk (m 2 ) φ(m 1 ) = 1, λαk (m 1 )λαk (m 2 )∗ φ(m 1 ) = −1,

(2.3)

where φ : M → Z2 = {±1} is an indicator for unitary/antiunitary symmetry. Next, we consider a more general element l ∈ M. A sewing matrix u αk on M, corresponding to λαk , can be defined by † lcαi (k)l −1 =



cα† j (l k)[u αk (l)] ji ,

(2.4)

j

where we note that the momentum l k is not equivalent to k for a general l ∈ M. u αk also satisfies the following equation:  ωin (l1 , l2 )u αk (l1l2 )

=

u lα2 k (l1 )u αk (l2 ) φ(l1 ) = 1, u lα2 k (l1 )u αk (l2 )∗ φ(l1 ) = −1.

(2.5)

Especially, u αk (m) = λαk (m) for m ∈ Mk . The factor system satisfies the following 2-cocycle condition, ωin (l1 , l2 )ωin (l1l2 , l3 ) = ωin (l1 , l2 l3 )ωin (l2 , l3 )φ(l1 ) ,

(2.6)

which is easily derived from the associativity of product of representation matrices. For easy treatment of the sewing matrix u αk (and the small corepresentation λαk ) of the magnetic space group M with infinite number of elements, it is useful to consider the finite factor group M/T. Then, the corresponding sewing matrix u¯ αk on M/T is introduced by ¯ k = Mk /T while jz is a basis speaking, α is a double-valued IR of the finite group M of the continuous group. Therefore there is no one-to-one correspondence between α and jz in some cases. In the C2v symmetry, for example, the IR α = 1/2 includes all normal Bloch states with half-integer total angular momentum jz = ±1/2, ±3/2, ±5/2, . . . . In this thesis, however, we ¯ k = α, as the angular-momentum represent jz with minimum absolute value, which satisfies jz ↓ M counterpart of the IR α.

2 Strictly

2.1 Preparation

13

¯ u αk (l) = e−il k·τ l u¯ αk (l),

(2.7)

where l¯ is a representative in M/T for l ∈ M, and τ l = t l − t l¯,

(2.8)

is a Bravais lattice translation in T corresponding to l. Substituting (2.7) into (2.5), we obtain  u¯ lα2 k (l1 )u¯ αk (l2 ) φ(l1 ) = 1, (2.9) ωin (l1 , l2 )e−il1 l2 k·ν(l1 ,l2 ) u¯ αk (l1l2 ) = u¯ lα2 k (l1 )u¯ αk (l2 )∗ φ(l1 ) = −1, ν(l1 , l2 ) = τ l1 l2 − τ l1 − pl1 τ l2 = −[t l1 l2 − t l1 − pl1 t l2 ].

(2.10)

In Eq. (2.9), the nonsymmorphic part of the factor system3 is described by a Bravais lattice translation ν(l1 , l2 ), namely k (l1 , l2 ) = e−i k·ν(l1 ,l2 ) , ωns

(2.11)

k k (l1 , l2 ) = ωns (l1 , l2 ). which depends only on the corresponding representatives: ωns k k k ¯ ¯ Especially, an IR λα of the magnetic little cogroup M = M /T, which corresponds to the small corepresentation λαk , satisfies the following equations:

¯ λαk (m) = e−i k·τ m λ¯ αk (m),  k λ¯ α (m 1 )λ¯ αk (m 2 ) φ(m 1 ) = 1, k (m 1 , m 2 )λ¯ αk (m 1 m 2 ) = ωin (m 1 , m 2 )ωns λ¯ αk (m 1 )λ¯ αk (m 2 )∗ φ(m 1 ) = −1,

(2.12) (2.13)

¯ k for m ∈ Mk . where m¯ is a representative in M In a similar way, a projective representation γ¯αk of the unitary little cogroup G¯ k , which corresponds to a small representation γαk of the little group G k , is naturally introduced. Supposing that a concrete expression of the representation γ¯αk is given, we show a construction method of the corresponding corepresentation λ¯ αk . The problem is whether a Kramers degeneracy occurs or not when the unitary group G¯ k is changed ¯ k by adding antiunitary operators. We can systematically solve the problem with to M k (l1 , l2 )} [16, 17], by using the appropriate factor system {ω k (l1 , l2 ) = ωin (l1 , l2 )ωns the Wigner criterion (Herring test) [17–21]:

3 The

factor system also satisfies the 2-cocycle condition: l1 l2 l3 k l1 l2 l3 k l1 l2 l3 k l2 l3 k ωns (l1 , l2 )ωns (l1 l2 , l3 ) = ωns (l1 , l2 l3 )ωns (l2 , l3 )φ(l1 ) .

14

2 Method

Table 2.1 Notations of groups and representations used in the thesis. The terminologies in the first column are associated with an unitary group, while those in the third column are with a nonunitary group including antiunitary operators. In the table, we adopt the terminologies of Ref. [17]. Reprinted with permission from Ref. [15]. Copyright © 2019 by the American Physical Society Terminology Notation Terminology Notation Definition Space group

G

Magnetic space group

M

Little group

Gk

Mk

Small representation Little cogroup

γαk

Magnetic little group Small corepresentation Magnetic little cogroup

N/A

γ¯αk

N/A

λ¯ αk

G¯ k

WαT

λαk ¯k M

Whole crystal symmetry of the system Stabilizer of k IR of (magnetic) little group Factor group of (magnetic) little group by T IR of (magnetic) little cogroup

⎧ ⎪ (a), ⎨1 1 ≡ ω k (Tg, ¯ Tg)χ ¯ [γ¯αk ((Tg)2 )] = −1 (b), ⎪ |G¯ k | ¯ k ⎩ g∈ ¯ G 0 (c), 

(2.14)

where χ is a character of the representation, T ≡ T I is a pseudo-TRS operator preserving k. In the (b) and (c) cases, the degeneracy (dimension) of λ¯ αk is twice as ¯ gives the same representation as γ¯αk (g) ¯ for g¯ ∈ G¯ k in much as that of γ¯αk , while λ¯ αk (g) the (a) case (for details, see Appendix 1). In the above discussion, we have introduced a lot of notations for groups and representations. For the avoidance of confusion, we summarize the notations in Table 2.1.

2.2 Group-Theoretical Classification of Superconducting Gap In this section, we introduce the group-theoretical classification of the superconducting gap on high-symmetry points in the BZ [1–9]. As seen in Sect. 2.1, we can obtain the small corepresentation λαk corresponding to the normal Bloch state on a high-symmetry k point. In the superconducting state, zero center-of-mass momentum Cooper pairs have to be formed between two degenerate states present at k and −k in the same band when we adopt the weak-coupling BCS theory. Then, the two states should be connected by the IS I and/or the TRS T , except for an accidentally degenerate case. As a result, the representation Pαk of the Cooper pair wave function can be constructed from the representations of the Bloch state λαk .

2.2 Group-Theoretical Classification of Superconducting Gap

15

Next, we calculate the (single-valued) representation Pαk of the Cooper pair wave function. Taking into account the antisymmetry of the Cooper pairs and the degeneracy of the two states, Pαk is naively regarded as an antisymmetrized Krok k necker square [17, 22] of the induced representation λαk ↑ Mpair . Here, Mpair ≡ k k k M + IM is the group to which the representation Pα of the pair wave function belongs. As is the case for the sewing matrix, we should consider a factor group k by T for easy treatment; namely, Pαk (m) = P¯αk (m), ¯ where m = mt ¯ for of Mpair k ¯ k ≡ Mk /T, and t ∈ T. The representation P¯αk of M ¯ k is m ∈ Mpair , m¯ ∈ M pair pair pair concerned with a double coset decomposition in the field of mathematics, and is systematically calculated by the following Mackey–Bradley theorem [17, 22, 23]. Theorem 2.1 (Mackey–Bradley theorem) Let λ¯ αk a projective corepresentation with ¯ k , which is a subgroup of a factor system ω k (l1 , l2 ) of the magnetic little cogroup M k ¯ Mpair such that the double coset decomposition 

k ¯ pair M =

¯ k dn M ¯ k, M

(2.15)

n=0,1

¯ be is satisfied. We choose d0 = E and d1 = I without loss of generality. Let N¯ αk (m) ¯ k . Then there exists an extension P¯αk of ¯ × λ¯ αk (I −1 mI) ¯ of M the representation λ¯ αk (m) k ¯ pair such that [ P¯αk (I)]i j,i  j  = −[λ¯ αk (E)]i  j [λ¯ αk (E)] j  i . Finally, N¯ αk into the group M an antisymmetrized Kronecker square of the induced representations satisfies the following equation, 

 k k k ¯ pair ¯ pair ¯ pair ) × (λ¯ αk ↑ M ) = λ¯ αk × λ¯ αk ↑ M + P¯αk , (λ¯ αk ↑ M

(2.16)

where the second term P¯αk in the right-hand side gives a representation of Cooper pair wave function. The characters of P¯αk are calculated by ¯ ω k (I, g) χ [λ¯ αk (g)] ¯ = k ¯ 2, χ [ P¯αk (g)] ω (g, ¯ I) ¯ k (g, ¯ g) ¯ ω k (I, g)ω χ [λ¯ αk (g 2 )], χ [ P¯αk (I g)] ¯ =− k ω (g, ¯ I)

(2.17a) (2.17b)

for g¯ ∈ G¯ k . Proof Here we prove Eq. (2.17) in the context of the Cooper pair wave function (for purely mathematical description, see Refs. [17, 22, 23]). First, we introduce a creation operator of a normal Bloch state in a band-based representation denoted by † ¯ k and its basis, cαi (k), where α and i are the IR of the magnetic little cogroup M respectively. This operator is transformed as † (k)m¯ −1 = mc ¯ αi

 i

† ¯ k ¯ i i , cαi  (k)[λα (m)]

(2.18)

16

2 Method

¯ k . Then, we define the Cooper pair wave function by m¯ ∈ M † (k) · Icα† j (k)I, [α (k)]i j ≡ cαi

(2.19)

where we note that a intra-band order parameter is assumed by considering a pair between two Bloch states belonging to the same IR α. Using Eq. (2.18), the Cooper pair wave function is transformed by g¯ ∈ G¯ k as follows4 : † † ¯ αi (k)g¯ −1 · I(I gI)c ¯ ¯ −1 I g[ ¯ α (k)]i j g¯ −1 = gc α j (k)(I gI) ⎧ ⎫ ⎨ ω k (I, g) ⎬  †  ¯ † k ¯  = cαi  (k)[λ¯ αk (g)] ¯ i i · I c (k)[ λ ( g)] ¯ I  j j α ⎩ ω k (g, ⎭ ¯ I) j  α j i

=

 i , j 

=



† † {cαi  (k) · Icα j  (k)I}

[α (k)]i  j 

i , j 

¯ k ω k (I, g) [λ¯ (g)] ¯ i  i [λ¯ αk (g)] ¯ j j k ω (g, ¯ I) α

ω k (I, g) ¯ k [λ¯ (g)] ¯ i  i [λ¯ αk (g)] ¯ j j . ω k (g, ¯ I) α

(2.20)

On the other hand, iαj (k) is transformed by I g¯ ∈ I G¯ k as † † ¯ −1 = −(I g)Ic ¯ ¯ −1 (I g)[ ¯ α (k)]i j (I g) α j (k)I · cαi (k)(I g) † † = −(I gI)c ¯ ¯ −1 · I{gc ¯ αi (k)g¯ −1 }I α j (k)(I gI)  † ¯ k ω k (I, g) [λ¯ α (g)] {cαi  (k) · Icα† j  (k)I} k ¯ i  j [λ¯ αk (g)] ¯ j i =− ω ( g, ¯ I) i , j 

=−



[α (k)]i  j 

i , j 

4

ω k (I, g) ¯ k [λ¯ (g)] ¯ i  j [λ¯ αk (g)] ¯ j i , k ω (g, ¯ I) α

Here, (I g¯ I )cα† j (k)(I g¯ I )−1 =



(I g)c ¯ α† j1 (I k)(I g) ¯ −1 [u¯ αk (I )] j1 j

j1

=

 j1 , j2

=

k I cα j2 (g¯ I k)I −1 [u¯ I ¯ j2 j1 [u¯ αk (I )] j1 j α (g)]



j1 , j2 , j3

†

k cα† j3 (I g¯ I k)[u¯ αg¯ I k (I )] j3 j2 [u¯ I ¯ j2 j1 [u¯ αk (I )] j1 j α (g)]

= ωI g¯ I k (I , g)ω ¯ I g¯ I k (I g, ¯ I)



cα† j3 (I g¯ I k)[u¯ αk (I g I )] j3 j

j3

¯  † ω k (I , g) = k c  (k)[λ¯ αk (g)] ¯ j j , ω (g, ¯ I)  αj j

where we use I g¯ I ∈ G k and I g I = g¯ in the last equal.

(2.21)

2.2 Group-Theoretical Classification of Superconducting Gap

17

where we use the anticommutation relation of fermions. From the above calculations, we obtain the representation of the Cooper pair wave function P¯αk : ¯ k ω k (I, g) [ P¯αk (g)] ¯ i j,i  j  = k ¯ i  i [λ¯ αk (g)] ¯ j j , [λ¯ (g)] ω (g, ¯ I) α ¯ k ω k (I, g) [λ¯ (g)] [ P¯αk (I g)] ¯ i j,i  j  = − k ¯ i  j [λ¯ αk (g)] ¯ j i . ω (g, ¯ I) α

(2.22a) (2.22b)

Therefore, the character of Pαk is given by χ [ P¯αk (g)] ¯ =

 ω k (I, g) ¯ ij

ω k (g, ¯ I)

¯ ii [λ¯ αk (g)] ¯ jj [λ¯ αk (g)]

¯ ω k (I, g) χ [λ¯ αk (g)] = k ¯ 2, ω (g, ¯ I)  ω k (I, g) ¯ k [λ¯ α (g)] χ [ P¯αk (I g)] ¯ =− ¯ i j [λ¯ αk (g)] ¯ ji k ω ( g, ¯ I) ij =−

¯ k (g, ¯ g) ¯ ω k (I, g)ω χ [λ¯ αk (g 2 )]. ω k (g, ¯ I)

(2.23a)

(2.23b)

Equations (2.23a) and (2.23b) are nothing but the Mackey–Bradley theorem.  k ¯ Finally, we decompose Pα into single-valued IRs of the (magnetic) point group ¯ k . The gap functions should be zero, and thus, the gap nodes appear, if the M pair corresponding IRs do not exist in the results of reductions [1, 2, 24]. Otherwise, the superconducting gap opens in general. Therefore the representation P¯αk of pair wave function tells us the presence or absence of superconducting gap nodes.

2.3 Topological Classification of Superconducting Gap In this section, we consider the topological classification of the superconducting gap on high-symmetry points (mirror plane or rotation axis) in the BZ. Before going to the main gap classification theory, let us overview the general framework of the topological classification. The topological classification is based on a recently developed classification theory of topological insulators/superconductors. Note that a topological insulator (superconductor) means, in a usual sense, an insulator (superconductor) where the wavefunctions of electrons (Bogoliubov quasiparticles) below the bandgap has a nontrivial topological number. The topological number is defined for the (BdG) Hamiltonian on the BZ, which is represented by a d-dimensional torus. In the present condensed matter physics, the topological periodic table (Table 2.2), which is a comprehensive classification table based on the onsite symmetries, namely TRS (T ),

18

2 Method

Table 2.2 Correspondence between the AZ classes and the onsite symmetries: TRS (T ), PHS (C ) and CS ( ) [27–29]. The fifth–eighth columns represent topological classification results of a Hamiltonian defined on a d-dimensional torus for each AZ class. Classifications in the complex classes (A, AIII) and those in the real classes (AI, . . . , CI) are respectively periodic, which is called Bott periodicity T C AZ class d = 0 d=1 d=2 d=3 0 0 +1 +1 0 −1 −1 −1 0 +1

0 0 0 +1 +1 +1 0 −1 −1 −1

0 1 0 1 0 1 0 1 0 1

A AIII AI BDI D DIII AII CII C CI

Z 0 Z Z2 Z2 0 2Z 0 0 0

0 Z 0 Z Z2 Z2 0 2Z 0 0

Z 0 0 0 Z Z2 Z2 0 2Z 0

0 Z 0 0 0 Z Z2 Z2 0 2Z

PHS (C), and CS ( ), is widely spread. The table is based on the tenfold AZ classes [25, 26], which are classified by four types of algebraic structure: 0, Z, 2Z, and Z2 . Now we apply the above classification theory of topological insulators or superconductors to the superconducting gap classification [12]. Note that a bandgap cannot exist in the whole d-dimensional BZ in nodal superconductors. Therefore, we consider a p-dimensional sphere surrounding the node, instead of the d-dimensional torus. p + 1 is called a codimension, which shows a difference between the dimension of the space and that of the node: for example, when a (1D) line node exists in a 3D space, the codimension is 3 − 1 = 2, which results in p = 1. It is necessary to reconsider symmetry since the domain of a topological number is changed. On a p-dimensional sphere surrounding a superconducting node, symmetries preserving the position of the node is only allowed. In crystalline superconductors, the node should be fixed to high-symmetry k points (mirror plane or rotational axis) in the BZ. As mentioned in Sect. 2.1, a unitary little cogroup on such high-symmetry k points is denoted by G¯ k , and an IR of G¯ k by α, which corresponds to the total angular momentum of the normal Bloch state jz = ±1/2, ±3/2, . . . . Furthermore, although the TRS T , the PHS C, and the IS I themselves do not preserve the position of the node, the combined symmetries of them are allowed on any k point. We define the symmetries preserving any k point by pseudo-TRS T ≡ T I, pseudo-PHS C ≡ CI, and CS ≡ T C. Thus stabilizer of the high-symmetry k point is represented by the following group: ¯ k = G¯ k + TG¯ k + CG¯ k + G¯ k . G

(2.24)

2.3 Topological Classification of Superconducting Gap

19

¯k A topological number is defined by using the BdG Hamiltonian with the symmetry G on the high-symmetry point. Although the above classification theory of topological ¯ k , an insulators/superconductors is apparently not applicable under the symmetry G effective AZ (EAZ) class can be defined on the BdG Hamiltonian, using the following ¯ k , U(1)φ ), we execute the way. Then, using the factor system {ω k (m 1 , m 2 )} ∈ H 2 (G Wigner criteria [17–21] for T and C, WαT ≡

1 ¯ |G k |

WαC ≡

1 ¯ |G k |

⎧ ⎪ ⎨1,  k k 2 ω (Tg, ¯ Tg)χ ¯ [γ¯α ((Tg) ¯ )] = −1, ⎪ ⎩ g∈ ¯ G¯ k 0, ⎧ ⎪ ⎨1,  k k 2 ω (Cg, ¯ Cg)χ ¯ [γ¯α ((Cg) ¯ )] = −1, ⎪ ⎩ g∈ ¯ G¯ k 0,

(2.25)

(2.26)

and the orthogonality test [20, 21] for : Wα

 ω k (g, ¯ )∗ 1  1, k ∗ k ≡ χ [γ¯α ( −1 g ) ]χ [γ¯α (g)] ¯ = ¯ ∗ 0. |G¯ k | ¯ k ω k ( , −1 g )

(2.27)

g∈ ¯ G

† † In the above tests, we investigate orthogonality between {cαi (k)} and {acαi (k)a −1 } † k (a = T, C, or ), where cαi (k) is the ith basis of the IR γ¯α (for details, see Appendixes 1 and 2). From Eqs. (2.25)–(2.27), we obtain the set of (WαT , WαC , Wα ), which indicates an EAZ symmetry class of the BdG Hamiltonian on the high-symmetry points by using the knowledge of K theory [21, 25, 26]. Table 2.3 shows the correspondence between the set of (WαT , WαC , Wα ) and the EAZ symmetry class.

! Remark WαT = Wα = 0 in FM superconductors, since both the pseudo-TRS (T) and the CS ( ) are broken.

Furthermore, from the EAZ symmetry class, we can classify the IR at the k point into 0, Z, 2Z or Z2 (Table 2.3). In this context, (WαT , WαC , Wα ) gives a symmetrybased topological classification of the Hamiltonian at each k point on the plane (line). When the plane (line) intersects a normal-state FS, a node composed of the intersection line (point) on the plane (line) hosts a codimension 1, and therefore it is surrounded by a 0D sphere. In other words, superconducting gap nodes on the plane (line) are classified by a 0D topological number (see p = 0 in Table 2.3). When the classification is nontrivial (Z, 2Z, or Z2 ), the intersection leads to a node characterized by the topological invariant. Otherwise, a gap opens at the intersection line (point).

20

2 Method

Table 2.3 Correspondence table between the set of (WαT , WαC , Wα ) and EAZ symmetry classes in centrosymmetric systems [12, 15, 30, 31]. The fifth–seventh columns show topological classification results of the IR at the k point with the codimension p + 1. Note that the Bott periodicity of the real classes is reversed from that in Table 2.2, since T and C do not flip the momentum contrary to T and C . Adapted with permission from Ref. [15]. Copyright © 2019 by the American Physical Society T (WαT ) C (WαC ) (Wα ) EAZ class p=0 p=1 p=2 0 0 +1 +1 0 −1 −1 −1 0 +1

0 0 0 +1 +1 +1 0 −1 −1 −1

0 1 0 1 0 1 0 1 0 1

A AIII AI BDI D DIII AII CII C CI

Z 0 Z Z2 Z2 0 2Z 0 0 0

0 Z Z2 Z2 0 2Z 0 0 0 Z

Z 0 Z2 0 2Z 0 0 0 Z Z2

2.4 Example: Space Group P21 /m In this section, we consider a unitary space group G = P21 /m as a simple example, which helps us understand the terminologies in Table 2.1 and the superconducting gap classification methods in Sects. 2.2 and 2.3. In the following discussion, we treat a magnetic space group M = G + {T |0}G = P21 /m1 assuming the PM system.

2.4.1 Preparation Since the magnetic space group M has a mirror operator Mz = {Mz | 2zˆ }, we focus on mirror-invariant planes k z = 0 and k z = π in the BZ, where the (magnetic) little group is G k = T + Mz T, M = G + TG = T + Mz T + TT + Mz TT. k

k

k

(2.28) (2.29)

Therefore the (magnetic) little cogroup is G¯ k = G k /T = {E, Mz }, ¯ k = Mk /T = {E, Mz , T, Mz T}. M We find that the factor system is

(2.30) (2.31)

2.4 Example: Space Group P21 /m

21

ωin (Mz , Mz ) = −1, ωin (T, T) = −1, ωin (Mz , T) = ωin (T, Mz ), k ωns (Mz , Mz )

= 1,

k ωns (TMz , TMz )

=e . ik z

(2.32a) (2.32b)

k The projective IRs γ¯±1/2 of the little cogroup G¯ k , with the factor system ω (Mz , Mz ) = −1, are given by k

k k (E) = 1, γ¯±1/2 (Mz ) = ±i, γ¯±1/2

(2.33)

which corresponds to spin-up and spin-down states, respectively. For the spin-up one (α = +1/2), the Wigner criterion [17–21] for pseudo-TRS T is 1  k k ω (Tg, ¯ Tg)χ ¯ [γ¯+1/2 ((Tg) ¯ 2 )] 2 ¯k g∈ ¯ G  1 0 k z = 0, = (−1 + eikz ) = 2 −1 k z = π.

T = W+1/2

(2.34)

k ¯ k can be constructed of the magnetic little cogroup M Thus a representation λ¯ +1/2 as follows: m¯

E  Mz  T  Mz T  (2.35) 0 1 10 0 i +i 0 k λ¯ +1/2 (m) ¯ −1 0 01 ieikz 0 0 −ieikz

Note that, in this case, the equivalent projective representation is obtained even when we start from the spin-down state (α = −1/2); thus, we do not explicitly distinguish the two (spin-up and spin-down) states in the following discussions. By using the above representation, the small corepresentation of m ∈ Mk is given by k k (m) = e−i k·R λ¯ 1/2 (m), ¯ λ1/2

(2.36)

¯ k and TR = {E|R} ∈ T. where m = mT ¯ R with m¯ ∈ M

2.4.2 Group-Theoretical Gap Classification Now we execute the group-theoretical gap classification (Sect. 2.2) on the mirrorinvariant planes in the BZ. First of all, the factor system about the IS I is represented as follows: ωin (I, Mz ) = ωin (Mz , I) = 1, k ωns (I, Mz )

=e , ik z

k ωns (Mz , I)

= 1.

(2.37a) (2.37b)

22

2 Method

Using the normal-state representation [Eq. (2.35)] and the Mackey–Bradley theorem k of Cooper pair wave function is given [Eq. (2.17)], therefore, the representation P¯1/2 by ω k (I, E) k k (E)] = k (E)]2 = 4, χ [ P¯1/2 χ [λ¯ 1/2 ω (E, I) ω k (I, Mz ) k k χ [λ¯ 1/2 (Mz )] = k (Mz )]2 = −eikz (1 − eikz )2 , χ [ P¯1/2 ω (Mz , I) ω k (I, E)ω k (E, E) k k χ [λ¯ 1/2 χ [ P¯1/2 (I)] = − (E 2 )] = −2, ω k (E, I) ω k (I, Mz )ω k (Mz , Mz ) k k χ [λ¯ 1/2 χ [ P¯1/2 (S2 )] = − (M2z )] = 2eikz , ω k (Mz , I)

(2.38a) (2.38b) (2.38c) (2.38d)

k where S2 = {C2 | 2zˆ }. We can decompose the pair representation P¯1/2 into IRs of the k k ¯ ¯ + I G : point group C2h ∼ G =

g¯ E Mz I S2 Decomposition k z =0 ¯ ¯ 4 0 −2 2 A g + 2 Au + Bu P1/2 (g) k z =π (g) ¯ 4 4 −2 −2 A g + 3Bu P¯1/2

(2.39)

The decomposition shows the difference of superconducting gap structures between the BP k z = 0 and the ZF k z = π , which is attributed to the nonsymmorphic screw symmetry S2 ; for example, Au appears on k z = 0, while it is prohibited on k z = π . This means that, if the superconducting pairing symmetry belongs to the Au IR, line nodes emerge only on the k z = π plane. The result is nothing but a counterexample of Blount’s theorem introduced in Sect. 1.1.2.2. Such an unusual nodal structure is considered to appear in some superconductors, e.g., UPt3 [6, 7, 13, 32– 34], UCoGe [8], and CrAs [9].

2.4.3 Topological Gap Classification We here confirm the group-theoretical classification result in the previous section, in terms of topological classification based on the EAZ symmetry class (Sect. 2.3). k have been given by In this case, the little cogroup G¯ k and its IR matrices γ¯±1/2 k Eqs. (2.30) and (2.33), respectively. Here we choose the representation matrix γ¯+1/2 of the IR α = +1/2 and apply the topological classification introduced in Sect. 2.3.

2.4 Example: Space Group P21 /m

2.4.3.1

23

A g Gap Function

Now we consider the A g pair wave function of the point group C2h . According to the group-theoretical gap classification, this function opens a superconducting gap on the planes [see Eq. (2.39)]. Before going to the tests for the EAZ symmetry class, we prepare some relationships among the intrinsic symmetries. For the A g pairing, we get the following factor system: ωin (C, C) = 1, ωin (C, Mz ) = ωin (Mz , C), k ωns (CMz , CMz )

=e . ik z

(2.40a) (2.40b)

Therefore5 ωin ( , Mz ) = ωin (Mz , ),

(2.41a)

k k ωns ( , Mz ) = ωns (Mz , ) = 1.

(2.41b)

k Next, we apply the topological classification to the normal Bloch state γ¯+1/2 and the A g pair wave function. Using the above factor system, the Wigner criteria for T and C and the orthogonality test for are given by



T W+1/2 C W+1/2

W+1/2

0 k z = 0, (2.42) −1 k z = π,  1 k k k ω (C, C)χ [γ¯+1/2 = (C2 )] + ω k (CMz , CMz )χ [γ¯+1/2 ((CMz )2 )] 2  1 0 k z = 0, = (+1 − eikz ) = (2.43) 2 +1 k z = π,  ω k (E, ) 1 k k χ [γ¯+1/2 = (E)∗ ]χ [γ¯+1/2 (E)] 2 ω k ( , −1 E )  ω k (Mz , ) k ∗ k −1 + k χ [γ¯+1/2 ( Mz ) ]χ [γ¯+1/2 (Mz )] ω ( , −1 Mz ) 1 (2.44) = (1 + 1) = 1 (k z = 0, π ). 2 =

5

Taking into account the factor system in Eqs. (2.32) and (2.40), we can easily derive it by the following commutation relation: [u( ), u(Mz )] = ωin (T, C)−1 {u(T)u(C)∗ u(Mz ) − u(Mz )u(T)u(C)∗ } = ωin (T, C)−1 {u(T)u(C)∗ u(Mz ) − u(T)u(Mz )∗ u(C)∗ } = ωin (T, C)−1 {u(T)u(C)∗ u(Mz ) − u(T)u(C)∗ u(Mz )} = 0, where we use = T C = TC.

24 Fig. 2.1 Schematic picture of the BdG Hamiltonian with the A g pair wave function on mirror-invariant planes a k z = 0 and b k z = π . The red-framed spaces for a and b belong to the EAZ class AIII and DIII, respectively. a is reprinted with permission from Ref. [15]. Copyright © 2019 by the American Physical Society

2 Method

(a)

(b)

EAZ class: AIII

EAZ class: DIII

Hole

Particle

Thus, according to Table 2.3, the classifying space is identified as the EAZ symmetry class AIII (DIII) on the k z = 0 (k z = π ) plane. Since the both classes are classified into 0, the gap classification is topologically trivial. This means that the A g pair wave function opens a gap on the mirror-invariant planes, which is consistent with the group-theoretical classification. Here we explain the meaning of the EAZ symmetry class. Figure 2.1 represents a schematic picture of the BdG Hamiltonian with the A g pair wave function. First, we see the classification on the BP k z = 0 (Fig. 2.1a). In the above discussion, we started k of the IR α = +1/2, which corresponds to the from the representation matrix γ¯+1/2 normal Bloch state (the lower left particle in Fig. 2.1a). The Wigner criterion for T = 0 [Eq. (2.42)], which indicates that the pseudo-TRS operator T results in W+1/2 T gives a basis of the nonequivalent IR (see Appendix 1). Therefore the lower left particle in Fig. 2.1a is mapped by T to the lower right particle, which belongs to the C = 0 [Eq. (2.43)], the lower left particle other IR α = −1/2. Similarly, since W+1/2 is mapped by C to the upper right hole. On the other hand, since the orthogonality = 1 [Eq. (2.44)], the CS gives the basis of the equivalent IR test leads to W+1/2 (see Appendix 2). Thus the lower left particle in Fig. 2.1a is mapped by to the upper left hole, which belongs to the same IR α = +1/2. For the above reason, the Hamiltonian in the space of α = +1/2 (the red frame in Fig. 2.1a) has only the CS , which indicates the AZ class AIII [25, 26]. On the ZF k z = π , on the other hand, the situation is changed (Fig. 2.1b). In this T C = −1 [Eq. (2.42)] and W+1/2 = +1 case, the Wigner criteria for T and C are W+1/2 [Eq. (2.43)], respectively. They indicate that each of T and C gives a basis of the =1 equivalent IR +1/2, and as a result, the CS also gives the same IR: W+1/2 [Eq. (2.44)]. Note that the mapped basis by the pseudo-TRS T is linearly independent T = −1 (see Appendix 1). Therefore the of the original basis, due to the result W+1/2 lower particle in Fig. 2.1b is mapped, by the any symmetry in T, C, and , to the particle and holes belonging to the same IR α = +1/2. That is why the Hamiltonian in the space of α = +1/2 (the red frame in Fig. 2.1b) is classified into the AZ class DIII [25, 26]. Furthermore, we see an intuitive understanding of the gap opening in the A g symmetry. For simplicity, we only treat the BP k z = 0 case in the discussion. Within

2.4 Example: Space Group P21 /m

25

the weak-coupling limit, i.e., for the negligibly small inter-band pairing, it is sufficient to discuss the single-band model. In this case, the BdG Hamiltonian HBdG (k) on the mirror plane is generally written as follows: 1 † C (k) Hˆ BdG (k)C(k), 2 † † † † C † (k) = (c+1/2 (k), Tc+1/2 (k)T−1 , Cc+1/2 (k)C−1 , c+1/2 (k) −1 ),

HBdG (k) =

(2.45) (2.46)

† where c+1/2 (k) is a creation operator of a jz = +1/2 Bloch state in a band-based k : representation, which is the basis of the IR γ¯+1/2 † † k gc+1/2 (k)g −1 = γ¯+1/2 (g)c+1/2 (k) for g ∈ G¯ k .

(2.47)

Hˆ BdG (k) is the matrix representation of HBdG (k): Hˆ BdG (k) =

 ξ(k)σ0 0 ψ(k)iσ y , (0 ψ(k)iσ y )† −ξ(k)σ0

(2.48)

where ξ(k) is the normal energy dispersion, and σi represents the Pauli matrix in pseudo-spin space. Here, the spin-singlet A g gap function is defined as   1 † † † † (k)( c+1/2 (k) −1 )† − Tc+1/2 (k)T−1 (Cc+1/2 (k)C−1 )† + H.c., 0 ψ(k) c+1/2 2 (2.49) where the magnitude of the gap function 0 is chosen as a real number without loss of generality. Due to the Mz symmetry, the BdG Hamiltonian matrix commutes with the mirror k (Mz ): reflection matrix Uˆ BdG k (Mz )] = 0. [ Hˆ BdG (k), Uˆ BdG

(2.50)

k Therefore Hˆ BdG (k) and Uˆ BdG (Mz ) are simultaneously block-diagonalized; namely, there exists a unitary matrix Vˆ such that

 Hˆ + (k) 0 Vˆ † , 0 Hˆ − (k) 

+i12 0 k Uˆ BdG (Mz ) = Vˆ Vˆ † , 0 −i12 Hˆ BdG (k) = Vˆ

(2.51) (2.52)

where we use ωkz =0 (Mz , Mz ) = −1. The block-diagonalized Hamiltonian H+ (k) and H− (k) are written as follows:

26

2 Method

(a)

(b)

Fig. 2.2 Band-theoretical picture of the BdG Hamiltonian with a the A g pair wave function and † is the abbreviated notation of b the Bg pair wave function on the mirror-invariant BP k z = 0. c+ † c+1/2 (k). The red points represent nodes on the plane. Reprinted with permission from Ref. [15]. Copyright © 2019 by the American Physical Society

1 † C ± (k) Hˆ ± (k)C ± (k), 2 

ξ(k) ±0 ψ(k) , Hˆ ± (k) = ±0 ψ(k)∗ −ξ(k) H± (k) =

† † C †+ (k) = (c+1/2 (k), c+1/2 (k) −1 ),

C †− (k)

=

† (Tc+1/2 (k)T−1 ,

† Cc+1/2 (k)C−1 ),

(2.53) (2.54) (2.55) (2.56)

k since does not change the eigenvalue of Uˆ BdG (Mz ), but T and C do that. Figure 2.2a schematically illustrates the band structures obtained by the BdG Hamiltonian (2.51). First, considering 0 → 0 limit (left panel in Fig. 2.2a), we get the particle band † † (k)c+1/2 (k) (ξ(k)Tc+1/2 (k)c+1/2 (k)T−1 ), ξ(k)c+1/2

(2.57)

and the hole band † † (k)c+1/2 (k) −1 (−ξ(k)Cc+1/2 (k)c+1/2 (k)C−1 ), − ξ(k) c+1/2

(2.58)

2.4 Example: Space Group P21 /m

27

in the eigenspace of the eigenvalue +i (−i). When the magnitude of the gap function 0 is finite, therefore, the A g pair wave function [Eq. (2.49)] can have finite offdiagonal components in each +i and −i eigenspace [see Eq. (2.54)]. These functions open the gaps at the zero energy (right panel in Fig. 2.2a). This fact indicates the existence of gap on the mirror-invariant plane.

2.4.3.2

B g Gap Function

Next, we investigate the Bg pair wave function on the mirror planes. Group-theoretical gap classification shows the emergence of line nodes on the planes [see Eq. (2.39)]. k and apply the topological classification. For the Bg We again choose the IR γ¯+1/2 pairing, the factor system is ωin (C, C) = 1, ωin (C, Mz ) = −ωin (Mz , C), k ωns (CMz , CMz )

=e . ik z

(2.59a) (2.59b)

Therefore, ωin ( , Mz ) = −ωin (Mz , ), k ωns ( , Mz )

=

k ωns (Mz , )

= 1.

(2.60a) (2.60b)

By using the above factor system, the Wigner criteria for T and C and the orthogonality test for are given by  T W+1/2

C W+1/2

=

0 k z = 0, −1 k z = π,

 1 +1 k z = 0, ik z = (+1 + e ) = 2 0 k z = π,

= W+1/2

1 (1 − 1) = 0 (k z = 0, π ). 2

(2.61) (2.62) (2.63)

According to Table 2.3, therefore, the classifying space is identified as the EAZ symmetry class D (AII) on the k z = 0 (k z = π ) plane. Since the class D (AII) is classified into Z2 (2Z), nodes emerge on the mirror-invariant planes by the Bg pair wave function when the FSs cross the planes. This node is topologically protected. Figure 2.3 represents the schematic picture of the BdG Hamiltonian with the Bg k pair wave function. On the BP k z = 0, the normal Bloch basis of γ¯+1/2 , namely, the lower left particle in Fig. 2.3a, is mapped by T to the lower right particle belonging to the other IR α = −1/2, because the Wigner criterion for the pseudo-TRS T is

28

2 Method

(a)

(b)

EAZ class: D

EAZ class: AII

Hole

Particle

Fig. 2.3 Schematic picture of the BdG Hamiltonian with the Bg pair wave function on mirrorinvariant planes a k z = 0 and b k z = π . The red-framed spaces for a and b belong to the EAZ class D and AII, respectively. a is reprinted with permission from Ref. [15]. Copyright © 2019 by the American Physical Society

T W+1/2 = 0 [Eq. (2.61)]. Similarly, since W+1/2 = 0 [Eq. (2.63)], the lower left particle is mapped by to the upper right hole. On the other hand, since the Wigner C = +1 [Eq. (2.62)], C gives the basis of the criterion for the pseudo-PHS is W+1/2 equivalent IR which does not generate an additional degeneracy (see Appendix 1). Thus the lower left particle in Fig. 2.3a is mapped by C to the upper left hole belonging to the same IR α = +1/2. For the above reason, the BdG Hamiltonian in the space of α = +1/2 (the red frame in Fig. 2.3a) has only the PHS C with C2 = +E, which indicates the AZ class D [25, 26]. On the ZF k z = π , the lower left particle in Fig. 2.3b is degenerated by T belongT = −1 ing to the equivalent IR α = +1/2, because of the Wigner criterion W+1/2 T [Eq. (2.61)]. On the other hand, since W+1/2 = W+1/2 = 0 [Eqs. (2.62) and (2.63)], the lower left particle is mapped by the pseudo-PHS C and the CS to the upper right holes in the nonequivalent IR α = −1/2. Therefore, the Hamiltonian in the space of α = +1/2 (the red frame in Fig. 2.3b) has only the TRS T with T2 = −E, which indicates the AZ class AII [25, 26]. Furthermore, we discuss an intuitive picture of the nodes appearing in the Bg symmetry for the BP k z = 0 case. As is the case for the A g IR, the BdG Hamiltonian k matrix Hˆ BdG (k) [Eq. (2.48)] and the mirror reflection matrix Uˆ BdG (Mz ) are simultaneously block-diagonalized on the mirror-invariant plane [Eqs. (2.51) and (2.52)]. For the Bg IR, the Hamiltonian blocks H+ (k) and H− (k) are written as follows:

1 † C (k) Hˆ ± (k)C ± (k), 2 ±

 ξ(k) 0 ˆ H± (k) = , 0 −ξ(k)

H± (k) =

(2.64) (2.65)

† † (k), Cc+1/2 (k)C−1 ), C †+ (k) = (c+1/2

(2.66)

† (Tc+1/2 (k)T−1 ,

(2.67)

C †− (k)

=

† c+1/2 (k) −1 ),

2.4 Example: Space Group P21 /m

29

k since C does not change the eigenvalue of Uˆ BdG (Mz ), but T and do that. Note ˆ that the off-diagonal components of H± (k) are zero because the spin-singlet Bg gap function has the same form as Eq. (2.49), which is not allowed in the equivalent eigenspace of ±i. In other words, the momentum dependence of the Bg gap function ψ(k) leads to the gap closing on the mirror plane, even when the magnitude 0 is finite. The band structures are schematically shown in Fig. 2.2b, which indicates the emergence of line nodes on the mirror plane k z = 0 (right panel in Fig. 2.2b). Finally, we comment on the Z2 classification of the gap node. As shown in Eq. (2.51), the BdG Hamiltonian matrix in this symmetry can be decomposed into the two matrices belonging to different eigenspaces of ±i: Hˆ + (k) ⊕ Hˆ − (k). Both matrices possess the pseudo-PHS C with C2 = +E. Therefore Hˆ ± (k) can be transformed into antisymmetric matrices by using the unitary part Uˆ C = Uˆ C,+ ⊕ Uˆ C,− of the pseudo-PHS C [35, 36]. Thus the nodes in each eigenspace are characterized by a Z2 number: (2.68) (−1)l± = sgn[i n Pf{Uˆ C,± Hˆ ± (k)}] ∈ Z2 ,

with n = dim( Hˆ ± )/2. This Z2 protection of nodes is the same as that of Bogoliubov FSs in even-parity chiral superconductors [36].

Appendix 1: Wigner Criterion In this Appendix, the method and meaning of the Wigner criterion, which is essential for the gap classification theory, are reviewed. For more detailed discussions of the criterion, see Refs. [17–21]. Let H be a unitary group and let α be a dα -dimensional projective IR of H , which has basis functions ψi (i = 1, 2, . . . , dα ). ψi is transformed by the operation h ∈ H as dα  hψi = ψ j [α (h)] ji , (2.69) j=1

where α is a representation matrix with the bases {ψi } of the IR α. Here we suppose a situation that a certain antiunitary operator a is added to the group: H + a H . Then, there exists a (projective) corepresentation Dα of H + a H , which is extended from the unitary IR α of H . The important question is whether the degeneracy (dimension) of the corepresentation is increased from dα or not. The problem can be solved by the following Wigner criterion [17–21]: ⎧ ⎪ (a), ⎨1  1 a 2 ω(ah, ah)χ [α ((ah) )] = −1 (b), Wα ≡ ⎪ |H | h∈H ⎩ 0 (c).

(2.70)

30

2 Method

Here, {ω(h 1 , h 2 )} ∈ H 2 (H + a H, U(1)φ ) is a factor system arising in the corepresentation  Dα (h 1 )Dα (h 2 ) (φ(h 1 ) = 1), ω(h 1 , h 2 )Dα (h 1 h 2 ) = (2.71) Dα (h 1 )Dα (h 2 )∗ (φ(h 1 ) = −1), where φ : H + a H → Z2 = {±1} is an indicator for unitary/antiunitary symmetry. The meanings of the cases (a), (b), and (c) are shown in the following. (a) There is no additional degeneracy in spite of the additional antiunitary operator a, because {ψi } and {aψi } are not independent. (b) The presence of the operator a invokes additional degeneracy, since {aψi } is linearly independent of {ψi } although they belong to the equivalent IR α. (c) The degeneracy is doubled by applying a, because {aψi } are bases of a representation α  inequivalent to α.

Example: C3 Symmetry As an example of the Wigner criterion, the rotational property of a basis of spin angular momentum ψ = |sz  is discussed. In a continuous space, |sz  is transformed by a θ rotation C(θ ) around the z axis as C(θ ) |sz  = eiθsz |sz  .

(2.72)

Now let us discuss threefold-rotational symmetric system H = {E, C3 , (C3 )2 }. Then the continuous rotational symmetry C(θ ) is restricted to discrete symmetries: θ = 0, 2π/3, and 4π/3. Next, we suppose the existence of the TRS T . The presence or absence of additional degeneracy for sz = +3/2 and +1/2 states in the group C3 + T C3 is investigated in the following. (b) sz = +3/2 case. The eigenvalue eiθsz (θ = 0, 2π/3, 4π/3) in Eq. (2.72), namely the 1D IR matrix +3/2 with the basis + 23 , is given by +3/2 (E) = 1, +3/2 (C3 ) = −1, +3/2 ((C3 )2 ) = 1.

(2.73)

Then, by adding the TRS T , the Wigner criterion leads to T = W+3/2

=

1 ω(T h, T h)χ [+3/2 ((T h)2 )] 3 h∈H 1 (−1 − 1 − 1) = −1. 3

(2.74)

2.4 Example: Space Group P21 /m

31

Note that the factor system is ω(T, h) = 1 for all h ∈ H , ω(T,  T)  = −1, and + 3 ∝ − 3 has (2.74) implies that T ω(C3 , C3 )ω((C3 )2 , C3 ) = −1. Equation 2 2   the same rotational property as + 23 : −3/2 (E) = 1, −3/2 (C3 ) = −1, −3/2 ((C3 )2 ) = 1,

(2.75)

therefore, −3/2 ≡ +3/2 .

(2.76)

    In other words, the bases + 23 and − 23 belong to the equivalent IR. However, the of the TRS T gives  rise  3presence   to additional degeneracy, since the basis − is linearly independent of + 3 . 2 2 (c) sz = +1/2 case.   From Eq. (2.72), the 1D IR matrix +1/2 with the basis + 21 is described by +1/2 (E) = 1, +1/2 (C3 ) = e+iπ/3 , +1/2 ((C3 )2 ) = e+i2π/3 .

(2.77)

The Wigner criterion for the TRS T is calculated by T = W+1/2

1 ω(T h, T h)χ [+1/2 ((T h)2 )] 3 h∈H

1 (−1 − ei2π/3 + eiπ/3 ) = 0. (2.78) 3  1  1 + ∝ − has the nonequivalent rotational Equation (2.78) implies that T 2 2  1  property to + 2 : =

−1/2 (E) = 1, −1/2 (C3 ) = e−iπ/3 , −1/2 ((C3 )2 ) = e−i2π/3 ,

(2.79)

therefore, −1/2 = +1/2 .

(2.80)

    Due to the bases + 21 and − 21 of nonequivalent IRs, the degeneracy is doubled by applying the TRS T .

Appendix 2: Orthogonality Test In the Appendix, the formulation of the orthogonality test for CS, which is used in the topological gap classification (Sect. 2.3), is explained. Let H be a unitary group and let α be a dα -dimensional projective IR of H , which has basis functions ψi (i = 1, 2, . . . , dα ). ψi is transformed by the operation h ∈ H

32

2 Method

as hψi =

dα 

ψ j [α (h)] ji ,

(2.81)

j=1

where α is a representation matrix with the bases {ψi } of the IR α. Then, we suppose the situation that the system possesses an additional CS : H + H . The orthogonality between {ψi } and { ψi } is investigated in the following. The basis ψi is transformed by h ∈ H as h( ψi ) = ( −1 h )ψi =

dα  ( ψ j )ω( −1 , h )ω(h, )[α ( −1 h )] ji , j=1

=

dα  ( ψ j ) j=1

ω(h, ) [α ( −1 h )] ji , ω( , −1 h )

(2.82)

where {ω(h 1 , h 2 )} ∈ H 2 (H + H, U(1)) is a factor system, and the 2-cocycle condition ω(h 1 , h 2 )ω(h 1 h 2 , h 3 ) = ω(h 1 , h 2 h 3 )ω(h 2 , h 3 ) for h 1 , h 2 , h 3 ∈ H + H, (2.83) is used. We remark that H is not changed under the CS: −1 H = H . In other words, ω(h, ) −1 h ). the representation matrix of H with the bases { ψi } is given by ω( , −1 h ) α ( Next, we recall the orthogonality relation between two IR matrices α and β [20, 21], 1  1 [α (h)]i j [β (h)∗ ]kl = δαβ δik δ jl . (2.84) |H | h∈H dα By taking i = j and k = l, we calculate the summation of Eq. (2.84) over i and k: dβ dα dα dα   1  δαβ  [α (h)]ii [β (h)∗ ]kk = δik , |H | h∈H i=1 dα i=1 k=1 k=1 1  ∴ χ [α (h)]χ [β (h)∗ ] = δαβ . |H | h∈H

(2.85)

Finally, Eqs. (2.82) and (2.85) lead to the orthogonality test between {ψi } and { ψi },

2.4 Example: Space Group P21 /m

1  ω(h, )∗ χ [α (h)]χ [α ( −1 h )∗ ] |H | h∈H ω( , −1 h )∗  1 ({ψi } and { ψi } are equivalent), = 0 ({ψi } and { ψi } are nonequivalent),

33

(2.86)

which is nothing but Eq. (2.27).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

Yarzhemsky VG, Murav’ev EN (1992) J Phys: Condens Matter 4(13):3525 Yarzhemsky VG (1998) Phys Status Solidi B 209(1):101 Yarzhemsky VG (2000) Int J Quant Chem 80(2):133 Yarzhemsky VG (2003) Conf AIP Proc 678(1):343 Yarzhemsky VG (2008) J Opt Adv Mater 10(7):1759 Micklitz T, Norman MR (2009) Phys Rev B 80:100506(R) Micklitz T, Norman MR (2017) Phys Rev B 95:024508 Nomoto T, Ikeda H (2017) J Phys Soc Jpn 86(2):023703 Micklitz T, Norman MR (2017) Phys Rev Lett 118:207001 Sumita S, Nomoto T, Yanase Y (2017) Phys Rev Lett 119:027001 Sumita S, Yanase Y (2018) Phys Rev B 97:134512 Kobayashi S, Shiozaki K, Tanaka Y, Sato M (2014) Phys Rev B 90:024516 Kobayashi S, Yanase Y, Sato M (2016) Phys Rev B 94:134512 Kobayashi S, Sumita S, Yanase Y, Sato M (2018) Phys Rev B 97:180504(R) Sumita S, Nomoto T, Shiozaki K, Yanase Y (2019) Phys Rev B 99:134513 Bradley CJ, Davies BL (1968) Rev Mod Phys 40:359 Bradley CJ, Cracknell AP (1972) The Mathematical theory of symmetry in solids. Oxford University Press, Oxford Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic, New York Herring C (1937) Phys Rev 52:361 Inui T, Tanabe Y, Onodera Y (1990) Group theory and its applications in physics, Springer Series in Solid-State Sciences, vol 78. Springer, Berlin Shiozaki K, Sato M, Gomi K, Atiyah-hirzebruch spectral sequence in band topology: general formalism and topological invariants for 230 space groups, arXiv:1802.06694 Bradley CJ, Davies BL (1970) J Math Phys 11:1536 Mackey GW (1953) Am J Math 75:387 Izyumov Y, Laptev V, Syromyatnikov V (1989) Int J Mod Phys B 3(9):1377 Zirnbauer MR (1996) J Math Phys 37(10):4986 Altland A, Zirnbauer MR (1997) Phys Rev B 55:1142 Schnyder AP, Ryu S, Furusaki A, Ludwig AWW (2008) Phys Rev B 78:195125 Kitaev A (2009) Conf AIP Proc 1134(1):22 Ryu S, Schnyder AP, Furusaki A, Ludwig AWW (2010) New J Phys 12(6):065010 Zhao YX, Schnyder AP, Wang ZD (2016) Phys Rev Lett 116:156402 Bzdušek Tcv , Sigrist M (2017) Phys Rev B 96:155105 Norman MR (1995) Phys Rev B 52:15093 Yanase Y (2016) Phys Rev B 94:174502 Nomoto T, Ikeda H (2016) Phys Rev Lett 117:217002 Ghosh P, Sau JD, Tewari S (2010) Das Sarma S 82:184525 Agterberg DF, Brydon PMR, Timm C (2017) Phys Rev Lett 118:127001

Chapter 3

Superconducting Gap Classification on High-Symmetry Planes

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes In this section, we completely classify symmetry-protected line nodes on highsymmetry planes [1–8], clarifying the condition for the presence or absence of line nodes protected by nonsymmorphic symmetry. Moreover, we show an additional constraint: line nodes (gap opening) characteristic to nonsymmorphic systems appear only on the ZF for 59 primitive or orthorhombic base-centered space groups. A gap classification table of the space groups is provided in Table 3.3, which helps us to search for further candidate superconductors with nonsymmorphic line nodes.

3.1.1 Setup First, we show constraints on crystal symmetry of the system where the formalism in Sect. 2.2 is applicable. As pointed out in the beginning of Sect. 2.1, the system is assumed to be invariant under the IS I, which ensures the degeneracy of two normal Bloch states at k and −k. The symmetry-protected line node may appear on high-symmetry k planes when a FS crosses the plane and a gap function vanishes there. Thus, we need to consider the k planes as high-symmetry k-points in order to discuss line nodes. Only an identity operation and a mirror (or glide) reflection are allowed as elements of the unitary little cogroup G¯ k at an arbitrary point on the high-symmetry k planes. For the above reasons, the point group of the system is assumed to contain at least C2h symmetry, which is generated by a spatial inversion and a mirror reflection. In other words, the space group of the system G has a subgroup H < G such that H/T ∼ = C2h . Taking translations into account, H is classified as follows: © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Sumita, Modern Classification Theory of Superconducting Gap Nodes, Springer Theses, https://doi.org/10.1007/978-981-33-4264-4_3

35

36

3 Superconducting Gap Classification on High-Symmetry Planes

(a)

Twofold axis

(b)

Fig. 3.1 a The coordinate r⊥ along the principal twofold axis, and the other coordinates r perpendicular to the principal axis. b An example of the AFM1 case. The red arrows illustrate magnetic moments on a square lattice, and the dashed line shows a magnetic unit cell. Reprinted with permission from Ref. [9]. Copyright © 2018 by the American Physical Society

H = {E|0}T + {M⊥ |t M }T + {I |0}T + {C2⊥ |t M }T, ⎧ 0 (RM) Rotation + Mirror, ⎪ ⎪ ⎪ ⎨t (RG) Rotation + Glide,  tM = ⎪ (SM) Screw + Mirror, t ⊥ ⎪ ⎪ ⎩ t  + t ⊥ (SG) Screw + Glide,

(3.1a)

(3.1b)

where t i = rˆi /2 (i =  or ⊥) are non-primitive translation vectors. E denotes an identity operation, M⊥ is a mirror reflection about the plane perpendicular to the r⊥ axis, and C2⊥ is a π -rotation around the r⊥ axis. Note that the direction of the principal (twofold) axis is represented by the symbol ⊥, while the other directions orthogonal to the principal axis are represented by the symbol  (Fig. 3.1a). In Eq. (3.1), the (RM) space group is symmorphic, while the other (RG), (SM), and (SG) space groups are nonsymmorphic. Next, we discuss magnetic (antiunitary) symmetry of the system. All the timereversal operation is forbidden when the system is FM. On the other hand, in the PM or AFM state, the system is invariant under the following time-reveal operation, T¯ = {T |t T }, ⎧ 0 ⎪ ⎪ ⎪ ⎨t  tT = ⎪ t⊥ ⎪ ⎪ ⎩ t + t⊥

(3.2a) (PM), (AFM1), (AFM2), (AFM3).

(3.2b)

The pure TRS is allowed in the (PM) state, while the system is invariant under the successive operations of time-reversal and non-primitive translation in the (AFM1)– (AFM3) states. For example, a magnetic structure of the (AFM1) state is shown

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes

37

in Fig. 3.1b: although magnetic moments (red arrows) flip under the time-reversal operation, the magnetic structure recovers after a half-translation t  = rˆ /2.

3.1.2 Gap Classification A magnetic space group of the superconductor is constructed by combining the unitary space group (3.1) and the antiunitary operators (3.2): M = H (FM) or M = H + T¯ H (PM or AFM). Based on the magnetic space group, the group-theoretical gap classification method introduced in Sect. 2.2 is applied to high-symmetry planes in the BZ: a BP at k⊥ = 0 and a ZF at k⊥ = π (lattice constants are set to 1 here). The obtained results are summarized in Table 3.1. In this table, representations of ¯k ∼ superconducting gap functions are classified by IRs of the point group M pair = C 2h . We also show candidate materials for unusual gap structures in Table 3.2, one of which is Sr2 IrO4 [10] discussed in Sect. 3.3.

Table 3.1 Group-theoretical classification of superconducting gap on high-symmetry k planes. H and T¯ specify a magnetic space group of the superconductor [see Eqs. (3.1) and (3.2)]. The representations of Cooper pairs allowed on the high-symmetry k planes (BP and ZF) are shown. Adapted with permission from Refs. [9, 11]. Copyright © 2018 by the American Physical Society Cases H T¯ Key ingredients BP (k⊥ = 0) ZF (k⊥ = π ) (1a) (1b) (2a)

(RM), (RG) (SM), (SG) (RM), (RG)

N/A N/A (PM), (AFM1)

(2b)

(RM), (RG)

(AFM2), (AFM3)

(2c)

(SM), (SG)

(PM), (AFM1)

(2d)

(SM), (SG)

(AFM2), (AFM3)

[t M ]⊥ = 0 [t M ]⊥ = 0 [t M ]⊥ = 0, [t T ]⊥ = 0 [t M ]⊥ = 0, [t T ]⊥ = 0 [t M ]⊥ = 0, [t T ]⊥ = 0 [t M ]⊥ = 0, [t T ]⊥ = 0

Au Au Au Bu Ag + 2 Au + Bu Ag + 2 Au + Bu Ag + 2 Au + Bu Bg + 3Au Ag + 2 Au + Bu Ag + 3Bu Ag + 2 Au + Bu Bg + Au + 2Bu

Table 3.2 Candidate materials with unusual gap structures on a high-symmetry ZF. The cases (1b), (2b), (2c), and (2d) correspond to those in Table 3.1. Adapted with permission from Ref. [9]. Copyright © 2018 by the American Physical Society Cases Materials (1b) (2b) (2c) (2d)

UCoGe (FM) [7], URhGe [7] Sr2 IrO4 (vertical) [10], UPt3 (AFM) [8] UPt3 (PM) [1–6], UCoGe (PM) [7], CrAs [8] UPd2 Al3 [7, 8, 12], UNi2 Al3 [8], Sr2 IrO4 (horizontal) [10]

38

3 Superconducting Gap Classification on High-Symmetry Planes

The classification of a superconducting gap on the BP in Table 3.1 is obviously independent of the presence or absence of nonsymmorphic symmetry, since the k is always trivial (1) on the BP. Thus the nonsymmorphic part of the factor system ωns classification result is consistent with the Sigrist–Ueda method. On the other hand, the representations allowed on the ZF differ from those on the BP in (1b), (2b), (2c), and (2d) cases. Then, the line nodes (or gap opening) protected by nonsymmorphic symmetry appear on the ZF. Such a situation is realized when the space group is (SM) or (SG), and/or the TRS is (AFM2) or (AFM3). In other words, when at least one of the translations [t M ]⊥ and [t T ]⊥ , where [·]⊥ means a vector component perpendicular to the mirror plane, is nonzero (non-primitive), unusual gap structures beyond the Sigrist–Ueda method are ensured by the nonsymmorphic symmetry. Results consistent with Table 3.1 have been recently shown by Micklitz and Norman [8]. Using a Clifford algebra technique, they also confirmed that the line nodes due to nonsymmorphic symmetry are protected by a Z topological number. A further discussion [11] about the topological stability of the line nodes and resulting surface states, namely Majorana flat bands, is given in Sect. 3.2.

3.1.3 Application to 59 Space Groups In the above discussion, we have shown the gap classification of line nodes on highsymmetry planes. Here we reconsider constraints on the crystal symmetry of the system. Space groups containing C2h symmetry are divided into four types: primitive, orthorhombic base-centered, body-centered, and face-centered space groups. All types of space groups have one or more mirror-invariant BPs (k⊥ = 0) in the BZ. On the other hand, corresponding ZFs (k⊥ = π ) exist only for primitive or orthorhombic base-centered space groups. Although some of body-centered or face-centered space groups also have mirror-invariant ZFs, all of them are k⊥ = 2π planes, where the gap classification gives the same result as that on BPs (k⊥ = 0 planes). Examples of mirror-invariant BPs and ZFs in the BZ of a primitive or orthorhombic base-centered Bravais lattice are illustrated in Fig. 3.2a–d. As shown in Table 3.1, unconventional line nodes (gap opening) characterized by nonsymmorphic symmetry appear on k⊥ = π ZFs. Therefore, we conclude that the nontrivial nonsymmorphic-symmetry-protected gap structures may appear not for a body-centered or face-centered Bravais lattice, but for a primitive or orthorhombic base-centered Bravais lattice. This additional constraint simplifies classification of space groups with respect to the superconducting gap structure. For the above reason, we classify here only primitive and orthorhombic basecentered space groups containing C2h symmetry, which may allow nontrivial gap structures due to nonsymmorphic symmetry. Intrinsic symmetry for each of BPs and ZFs (colored planes in Fig. 3.2a–d) is classified into (RM), (RG), (SM), or (SG) of Eq. (3.1). This enables us to determine a superconducting gap structure on the plane, using Table 3.1, when the magnetic symmetry is given. The results of the space group classification and the corresponding gap classification for PM superconductors

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes

(a)

39

(b)

(c)

(d)

Fig. 3.2 Mirror-invariant BPs and ZFs in the first BZ for a monoclinic primitive, b orthorhombic base-centered, c orthorhombic primitive/tetragonal primitive/cubic primitive, and d hexagonal primitive Bravais lattice. The blue and red planes represent BPs and ZFs, respectively. Reprinted with permission from Ref. [9]. Copyright © 2018 by the American Physical Society

are summarized in Tables 3.3a–e. Even for FM and AFM superconductors, we can straightforwardly elucidate the gap structure by combining Tables 3.1 and 3.3. Therefore, the group-theoretical classification on high-symmetry planes is completed. Finally, space groups of candidate materials for nonsymmorphic-symmetryprotected line nodes (gap opening) are listed in the following. • PM UPt3 [1–6]: G = P63 /mmc (Table 3.3d), T¯ = {T |0}; • PM UCoGe [7]: G = Pnma (Table 3.3b), T¯ = {T |0}; • FM UCoGe [7]:

G = P21 /c (Table 3.3a);

Short

Pmmm

Pnnn

Pccm

Pban

Pmma

Pnna

47

48

49

50

51

52

Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 3B1u + 3B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SG)

(RG)

y

z

(RG)

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

(RM)

y

z

Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 3B1u + 3B2u

(SM)

x

(RG)

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 2 Au + B1u + B2u + 2B3u

(RG)

z

x

Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

(RG)

y

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

(RM)

(RG)

z

x

Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 2 Au + B1u + 2B2u + B3u

(RG)

y

(RG)

Ag + B1g + 2 Au + 2B1u + B2u + B3u

(RG)

y

z

Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

(RG)

x

(RG)

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 2 Au + B1u + B2u + 2B3u

(RM)

z

x

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + 3Bu

Ag + 2 Au + Bu

Ag + 2 Au + Bu

(RM)

(SG)

Ag + 3Bu

Ag + 2 Au + Bu

(RM)

P21 /c

14

(RG)

Ag + 2 Au + Bu

k⊥ = π

Ag + 2 Au + Bu

Ag + 2 Au + Bu

k⊥ = 0

y

y

P2/c

13

(SM)

(RM)

Type

x

y

P21 /m

(b) Orthorhombic

y

y

P2/m

11

⊥

10

(a) Monoclinic

No.

Table 3.3 Gap classification for (a) monoclinic, (b) orthorhombic, (c) tetragonal, (d) hexagonal, and (e) cubic space groups. The table (b) shows results for primitive and base-centered Bravais lattices, while the other tables (a) and (c)–(e) show for a primitive Bravais lattice. The first and second columns represent the number and the short notation of space groups, respectively. The fourth column represents the type of the space groups in Eq. (3.1), corresponding to the direction (in the third column) of mirror-invariant BP and ZF. (see also Fig. 3.2a–d). Note that the direction “v” means x and y for the (c) tetragonal space groups, and [1−10], [120], and [210] for the (d) hexagonal ones. The fifth and sixth columns show the gap classification for PM superconductors on the BP k⊥ = 0 and the ZF k⊥ = π , respectively. Adapted with permission from Ref. [9]. Copyright © 2018 by the American Physical Society

40 3 Superconducting Gap Classification on High-Symmetry Planes

Short

Pmna

Pcca

Pbam

Pccn

Pbcm

Pnnm

Pmmn

Pbcn

No.

53

54

55

56

57

58

59

60

Ag + B2g + 3B1u + 3B3u

Ag + B2g + 3B1u + 3B3u

Ag + B3g + 3B1u + 3B2u Ag + B2g + 2 Au + B1u + 2B2u + B3u Ag + B1g + 3B2u + 3B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SG)

(RG)

(SG)

z

Ag + B2g + 3B1u + 3B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

(RG)

z

y

Ag + B3g + 3B1u + 3B2u

Ag + B2g + 2 Au + B1u + 2B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SM)

y

x

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

(SM)

x

(RM)

Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SG)

y

z

Ag + B1g + 3B2u + 3B3u Ag + B3g + 3B1u + 3B2u

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 2 Au + B1u + 2B2u + B3u

(SM)

z

(SG)

Ag + B2g + 3B1u + 3B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

(SG)

x

Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

(RG)

Ag + B2g + 3B1u + 3B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

(RG)

z

y

Ag + B3g + 3B1u + 3B2u

Ag + B2g + 2 Au + B1u + 2B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SG)

y

x

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

(SG)

x

(RM)

Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SG)

y

z

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 3B1u + 3B2u

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 2 Au + B1u + 2B2u + B3u

(RG)

z

(SG)

Ag + B2g + 2 Au + B1u + 2B2u + B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u

x

Ag + B3g + 3B1u + 3B2u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

(RG)

(SG)

z

(SG)

Ag + B2g + 2 Au + B1u + 2B2u + B3u Ag + B1g + 3B2u + 3B3u

Ag + B2g + 2 Au + B1u + 2B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

(RG)

y

y

Ag + B3g + 2 Au + B1u + B2u + 2B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u

x

k⊥ = π

k⊥ = 0

Type

(RM)

x

⊥

Table 3.3 (continued)

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes 41

Pbca

Pnma

Cmcm

Cmca

Cmmm

Cccm

Cmma

Ccca

61

62

63

64

65

66

67

68

Ag + B1g + 3B2u + 3B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

P4/mmm

P4/mcc

123

124

(RG)

(RM)

(RG)

z

v

(RM)

P42 /n

86

(RG)

v

z

P4/n

85

(RM)

(RM)

(RM)

z

P42 /m

(RG)

(RG)

(RM)

(RM)

z

z

z

P4/m

84

z

z

z

z

(SG)

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

Ag + Bg + 2 Au + 2Bu + E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

Ag + Bg + 2 Au + 2Bu + E u

Ag + Bg + 2 Au + 2Bu + E u

Ag + Bg + 2 Au + 2Bu + E u

Ag + Bg + 2 Au + 2Bu + E u

Ag + Bg + 2 Au + 2Bu + E u

Ag + Bg + 2 Au + 2Bu + E u

Ag + B1g + 3B2u + 3B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

z

Ag + Bg + 2 Au + 2Bu + E u

Ag + B2g + 3B1u + 3B3u Ag + B1g + 3B2u + 3B3u

(SG)

z

(SM)

Ag + B3g + 3B1u + 3B2u

Ag + B2g + 2 Au + B1u + 2B2u + B3u Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SM)

y

z

Ag + B1g + 3B2u + 3B3u

Ag + B1g + 2 Au + 2B1u + B2u + B3u

(SG)

z

Ag + B3g + 2 Au + B1u + B2u + 2B3u

(SG)

y

(SG)

Ag + B3g + 3B1u + 3B2u Ag + B2g + 3B1u + 3B3u

Ag + B3g + 2 Au + B1u + B2u + 2B3u Ag + B2g + 2 Au + B1u + 2B2u + B3u

x

k⊥ = π

k⊥ = 0

Type

(SG)

x

⊥

83

(c) Tetragonal

Short

No.

Table 3.3 (continued)

42 3 Superconducting Gap Classification on High-Symmetry Planes

Short

P4/nbm

P4/nnc

P4/mbm

P4/mnc

P4/nmm

P4/ncc

No.

125

126

127

128

129

130

(RG)

(SG)

v

(SM)

v

z

(RG)

(SG)

z

(RM)

z

(SG)

v

v

(RM)

(RG)

z

(RG)

z

v

(RG)

v

Type

(RG)

z

⊥

Table 3.3 (continued)

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 3A2u + 3B2u + 3E u

A1g + B1g + E g + 3A2u + 3B2u + 3E u A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 3A2u + 3B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 3A2u + 3B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

k⊥ = π

k⊥ = 0

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes 43

P42 /nbc

P42 /nnm

P42 /mbc

133

134

135

(RM)

(SG)

z

v

(RG)

v

(RG)

(RG)

v

z

(RG)

z

(RG)

v

P42 /mcm

132

(RM)

v

z

(RM)

z

P42 /mmc

131

Type

(RM)

⊥

Short

No.

Table 3.3 (continued)

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

k⊥ = 0

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 3A2u + 3B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

k⊥ = π

44 3 Superconducting Gap Classification on High-Symmetry Planes

(RM)

(RM)

P6/mmm

191

(RM)

(SM)

z

P63 /m

v

z

z

P6/m

(SG)

v

176

(RG)

z

175

(d) Hexagonal

P42 /ncm

(SM)

138

(RG)

P42 /nmc

137

z

v

v

(SG)

z

P42 /mnm

136

Type

(RM)

⊥

Short

No.

Table 3.3 (continued)

A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u

A1g + A2g + 2E 2g + 2 A1u + 2 A2u + B1u + B2u + 2E 1u + 4E 2u

A1g + A2g + 2E 2g + 2 A1u + 2 A2u + B1u + B2u + 2E 1u + 4E 2u

A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u

Ag + E 2g + 2 Au + Bu + E 1u + 2E 2u Ag + E 2g + 3Bu + 3E 1u

A1g + B1g + E g + 3A2u + 3B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 3A2u + 3B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 3A2u + 3B2u + 3E u

k⊥ = π

Ag + E 2g + 2 Au + Bu + E 1u + 2E 2u

Ag + E 2g + 2 Au + Bu + E 1u + 2E 2u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u

A1g + A2g + B1g + B2g + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u

k⊥ = 0

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes 45

P63 /mcm

P63 /mmc

193

194

224

223

222

221

205

201

200

x, y, z

x, y, z

¯ Pn 3m

x, y, z

x, y, z

x, y, z

x, y, z

(RG)

(RM)

(RG)

(RM)

(SG)

(RG)

(RM)

(RG)

v

x, y, z

(SM)

(RM)

z

(SM)

z

v

(RG)

v

Type

(RM)

z

⊥

¯ Pn 3n ¯ Pm 3n

Pa 3¯ ¯ Pm 3m

Pm 3¯ Pn 3¯

P6/mcc

192

(e) Cubic

Short

No.

Table 3.3 (continued)

A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u

A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u

A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u

Ag + E g + Tg + 6Tu

A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u A1g + A2g + 2E g + T1g + T2g + 2 A1u + 2 A2u + 4E u + 4T1u + 4T2u

Ag + E g + Tg + 2 Au + 2E u + 4Tu

Ag + E g + Tg + 2 Au + 2E u + 4Tu

Ag + E g + Tg + 2 Au + 2E u + 4Tu

Ag + E g + Tg + 2 Au + 2E u + 4Tu

A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u

A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u A1g + A2g + 2E 2g + 3B1u + 3B2u + 6E 1u

Ag + E g + Tg + 2 Au + 2E u + 4Tu

A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u A1g + A2g + 2E 2g + 2 A1u + 2 A2u + B1u + B2u + 2E 1u + 4E 2u A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u

A1g + A2g + 2E 2g + 3B1u + 3B2u + 6E 1u

A1g + A2g + 2E 2g + 2 A1u + 2 A2u + B1u + B2u + 2E 1u + 4E 2u A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u

A1g + A2g + 2E 2g + 2 A1u + 2 A2u + B1u + B2u + 2E 1u + 4E 2u A1g + B2g + E 1g + E 2g + 2 A1u + A2u + B1u + 2B2u + 3E 1u + 3E 2u

A1g + A2g + 2E 2g + 2 A1u + 2 A2u + B1u + B2u + 2E 1u + 4E 2u

k⊥ = π

k⊥ = 0

46 3 Superconducting Gap Classification on High-Symmetry Planes

3.1 Group-Theoretical Classification of Symmetry-Protected Line Nodes

47

• AFM UPd2 Al3 [7, 8, 12]: G = P21 /m (Table 3.3a), T¯ = {T |t ⊥ }; • AFM Sr2 IrO4 [10]: G = Pcca (Table 3.3b), T¯ = {T |t  + t ⊥ }.

3.2 Topological Classification of Symmetry-Protected Line Nodes and Majorana Flat Bands Let us now discuss the symmetry-protected line nodes from the viewpoint of topology. Such topological properties are important for identifying a bulk-boundary correspondence, i.e., the line-node-induced Majorana flat bands. In the following, we formulate the topology of line nodes on high-symmetry k planes. Although timereversal-symmetric superconductors are assumed in this section, the classification scheme can be straightforwardly applied to TRS-broken superconductors. The BdG Hamiltonian of a superconductor is represented by HBdG =

1 † C i (k)[ Hˆ BdG (k)]i j C j (k), 2

(3.3)

k,i, j

with C i† (k) = (ci† (k), ci (−k)). We note that the BdG Hamiltonian exhibits PHS: ¯ Hˆ BdG (k)∗ Uˆ BdG (C) ¯ † = − Hˆ BdG (−k), Uˆ BdG (C)

(3.4)

¯ = τx is an antiunitary operator, and τi (i = x, y, z) is a Pauli matrix where Uˆ BdG (C) in Nambu space. We choose a periodic Bloch basis so that Hˆ BdG (k) = Hˆ BdG (k + G), where G is a reciprocal lattice vector. First, we consider how symmetry operations affect the BdG Hamiltonian. The creation operator of an electron transforms under a unitary space group operation g ∈ G as  † c j (gk)[e−igk·τ g u¯ k (g)] ¯ ji , (3.5) gci† (k)g −1 = j

where we especially focus on C2h symmetry for the same reason mentioned in ¯ M ¯ ⊥, Sect. 3.1.1; namely, g¯ = {I |0}, {M⊥ |t M }, and {C2⊥ |t M }. We dub them as I, and C¯2⊥ , respectively. The BdG Hamiltonian with the C2h symmetry is invariant with respect to the symmetry operations, which are represented, in a matrix form, by k k (g) ¯ Hˆ BdG (k)Uˆ BdG (g) ¯ † = Hˆ BdG (gk), Uˆ BdG

(3.6)

48

3 Superconducting Gap Classification on High-Symmetry Planes

k with Uˆ BdG (g) ¯ = diag[u¯ k (g), ¯ ηg u¯ −k (g) ¯ ∗ ]. Here, ηg = ωin (C, g)/ωin (g, C) = ±1, which has the same sign as the character of g¯ in the IR of the order parameter. In addition, TRS T = {T |t T } acts on the creation operators as

which yields

T ci† (k)T −1 = c†j (−k)[ei k·τ T u¯ k (T¯ )] ji ,

(3.7)

k k (T¯ ) Hˆ BdG (k)∗ Uˆ BdG (T¯ )† = Hˆ BdG (−k), Uˆ BdG

(3.8)

k with Uˆ BdG (T¯ ) = diag[u¯ k (T¯ ), u¯ −k (T¯ )∗ ]. Next, we clarify the relations between the symmetry operations. On the mirror plane, the PHS operator satisfies −k k ¯ Uˆ BdG ¯ (g) ¯ ∗ = ηg Uˆ BdG (g) ¯ Uˆ BdG (C), Uˆ BdG (C)

(3.9)

k ¯ ⊥ ), Uˆ k (I), ¯ and Uˆ k (T¯ ) satisfy (M while Uˆ BdG BdG BdG k k k ¯ Uˆ BdG ¯ ⊥ ) = exp(−2i k · t M )Uˆ −k (M ¯ ⊥ )Uˆ BdG ¯ Uˆ BdG (I) (M (I), BdG −k k k ∗ k ¯ ⊥ ) = exp(−2i k⊥ · t T )Uˆ (M ¯ ⊥ )Uˆ BdG (T¯ ), Uˆ BdG (T¯ )Uˆ BdG (M BdG

(3.10a) (3.10b)

where we use ωin (I, M⊥ ) = ωin (M⊥ , I), ωin (T , M⊥ ) = ωin (M⊥ , T ), and the k (g1 , g2 ) defined in Eq. (2.11). nonsymmorphic part of factor system ωns In general, a line node appears either at (a) a general position or (b) a highsymmetry k plane. See Fig. 3.3a. For the both case, symmetries keeping any k point are intrinsic to the line node. Such symmetry operations are already given in Sect. 2.3: the pseudo-TRS T, the pseudo-PHS C, and the CS . The matrix representations of −k ¯ ˆ k ¯ ∗ K , Ck ≡ Uˆ BdG (C) ¯ Uˆ k (I) ¯ ∗ K , and  k ≡ (T )UBdG (I) the operators are Tk ≡ Uˆ BdG BdG −k ¯ ˆ ¯ ∗ , respectively. Here, K is the complex conjugate operator and i Uˆ BdG (T )UBdG (C) k 2 k 2 ηI (C ) = −(T ) = ( k )2 = 1. For the latter case, furthermore, the line node is k ¯ ⊥ ). Since (M fixed on a high-symmetry plane by the mirror reflection operation Uˆ BdG k 2 −2i k ·t  M ¯ [Uˆ BdG (M⊥ )] = −e , the sign of the squared operator may change at the ZF. k ¯ ⊥ ) which satisfies (M We conveniently fix the sign by defining Mk ≡ ei k ·t M Uˆ BdG k 2 (M ) = −1. Using Eqs. (3.9) and (3.10), we then obtain the following commutation relations, Ck Mk = e−2i k⊥ ·t M ηM⊥ Mk Ck ,  M =e k

k

−2i k⊥ ·t T

ηM⊥ M  , k

k

(3.11a) (3.11b)

while the commutation relation between Tk and Mk can be determined from Eq. (3.11). The right-hand sides of Eqs. (3.11a) and (3.11b) change their sign in between the BP and the ZF, when [t M ]⊥ and/or [t T ]⊥ are nonzero. This is obviously the same condition which we obtained in the group-theoretical classification (Sect. 3.1.2).

3.2 Topological Classification of Symmetry-Protected …

49

3.2.1 Line Node at a General Position As a vortex in the momentum space, the stability of a line node is ensured by a 1D topological number defined on a circle enclosing the line node. However, neither the pseudo-TRS nor pseudo-PHS operator gives a nonzero 1D topological number by itself. Thus the combination of the symmetries, namely the CS is required [13– 15]. Furthermore, since (Ck )2 = ηI , even-parity superconductors (ηI = 1) and oddparity ones (ηI = −1) belong to different EAZ classes DIII and CII, respectively. According to Table 2.3, only even-parity superconductors can support a non-zero 1D topological number. For even-parity superconductors, the 1D topological number in terms of  k can be defined on a loop l, Wl =

i 4π



d k · Tr[ k Hˆ BdG (k)−1 ∂ k Hˆ BdG (k)].

(3.12)

l

3.2.2 Line Node on a High-Symmetry Plane Let us move on to the other case, which is a main subject in this chapter: a line node on a high-symmetry k plane. We take Mk into account as a symmetry operation relevant to the line node, in addition to Ck , Tk , and  k . In this case, the line node separates the high-symmetry plane (2D BZ) into two disjoint regions, which are distinguished by a 0D topological number (see Fig. 3.3b). On the plane, the BdG Hamiltonian commutes with the mirror operator, and therefore splits into mirror sectors, Hˆ BdG → hˆ λ ⊕ hˆ −λ , where λ is an eigenvalue of Mk . Then,  k (Ck and Tk ) exists within the mirror sectors if it (anti)commutes with Mk . For instance, when [Ck , Mk ] = {Tk , Mk } = { k , Mk } = 0, the EAZ symmetry class of each mirror sector is the AII [(Tk )2 = −1], whose 0D topological number is 2Z. In a similar way, we have determined the possible 0D topological numbers in the mirror sectors

Fig. 3.3 a When a line node appears at a general position k0 , it is encircled by a loop l. b When a line node lies on a high-symmetry plane, two points kin and kout encircle it. Reprinted with permission from Ref. [11]. Copyright © 2018 by the American Physical Society

50

3 Superconducting Gap Classification on High-Symmetry Planes pq

Table 3.4 (a) Topology of the symmetry-protected line nodes, labeled by the symbol Mg(u) . The superscripts p and q are defined by p = exp(−2i k⊥ · t M )ηM⊥ and q = exp(−2i k⊥ · t T )ηM⊥ , respectively, while the subscript g (u) indicates even-parity (odd-parity) superconductivity. (b) The comparison between Table 3.1 and the 0D topological numbers, where Ag (Bu ) and Bg (Au ) on the ++ −− BP correspond to the Mg(u) and Mg(u) classes, respectively. Cases (2a)–(2d) correspond to those in Table 3.1. The labels (i)–(iii) indicate the superconductor class of the Majorana flat bands. Reprinted with permission from Ref. [11]. Copyright © 2018 by the American Physical Society (a) Topological No. 0D 1D

++

+−

0 2

2 2

−+

0 2

−−

++

+−

−+

2

0 0

2 0

0 0

ZF

BP 0 0 (i) 0 0

ZF 0 0 2 0

BP 0 (i) 0 0 0

2

−−

0 0

(b) Cases (2a) (2b) (2c) (2d)

BP 0 (iii) 0 0 (iii) 0

ZF 0 2 0 2

BP (ii) 2 (iii) 2 (ii) 2 (iii) 2

2

0 2 0

ZF 0 2 0 0

for all possible (anti)commutation relations. The results are summarized in Table 3.4. The obtained 0D topological numbers 2Z and Z2 of the line node in the mirror sector h λ can then be defined as Nˆ λ = n(k ˆ out )λ − n(k ˆ in )λ ,  Pf{hˆ λ (kout )L kout ,λ } νˆ λ (−1) = sgn , Pf{hˆ λ (kin )L kin ,λ }

(3.13) (3.14)

respectively, where n(k) ˆ λ is the number of occupied states with the mirror eigenvalue λ on a momentum k. (L k,λ is a sector-separated pseudo-PHS: Ck = (L k,λ ⊕ L k,−λ )K , and kin (kout ) is a momentum inside (outside) of the nodal loop. In the weak coupling limit, i.e., (k) → 0, Eqs. (3.13) and (3.14) are directly linked to the FS topology; when we define Nλ = n(kout )λ − n(kin )λ in the normal Hamiltonian to be the topological number of the FS in the mirror sector with the eigenvalue λ, Eqs. (3.13) and (3.14) are reduced to Nˆ λ = 2Nλ ,

(3.15)

(−1)νˆλ = (−1)Nλ ,

(3.16)

which implies that the nodal loop is only topologically stable if Nλ = 0. Now we elucidate the relationship between the group-theoretical and topological classifications. The IRs Ag , Au , Bg , and Bu correspond to the topological classifications labeled by Mg++ , Mu−− , Mg−− , and Mu++ on the BP, where the superscripts of

3.2 Topological Classification of Symmetry-Protected …

51

the symbol encode the commutation relations (3.11a) and (3.11b) and the subscript indicates even (g) or odd (u) parity [see also the caption of Table 3.4]. On the ZF, furthermore, the mirror symmetry classes depend on the symmorphic/nonsymmorphic properties, namely [t M ]⊥ and [t T ]⊥ due to Eq. (3.11). In Tables 3.1 and 3.4, we find a one-to-one correspondence, in which the absence of IRs coincides with the presence of the 0D topological numbers.

3.2.3 Possible Majorana Flat Bands Finally, we consider the connection between symmetry-protected line nodes and Majorana flat bands, which are characterized by 0D and 1D topological numbers, respectively. As discussed above, the 1D topological number (and corresponding Majorana flat bands) is allowed only for even-parity superconductors. In such evenparity superconductors, the 0D and 1D topological numbers are intrinsically related to each other: (3.17) |Nˆ λ | = |Wl |, (−1)νˆλ = (−1)Wl /2 , which implies that a symmetry-protected line node is always accompanied by the 1D topological number. Note that the 1D topological number survives as long as the CS is preserved even when the mirror symmetry is broken, which reflects the strong stability of the line nodes. According to Table 3.4 and Eq. (3.17), three classes about the stability of the line node are identified. A line node may be protected by (i) the 0D topological number in odd-parity superconductors with TRS (or a magnetic translation symmetry), or alternatively, by both the 0D and 1D topological numbers in even-parity superconductors with (ii) TRS or (iii) a magnetic translation symmetry. We immediately find that no Majorana flat band exists in class (i) superconductors since there is no 1D topological number in odd-parity superconductors. In class (ii) and (iii) superconductors, however, there is a chance to have Majorana flat bands due to the presence of the 1D topological number. In order to demonstrate the existence of Majorana flat bands, let us consider a system with an open boundary, e.g., the xi = 0 plane. Then, an Majorana flat band appears on the xi = 0 plane when the 1D topological number in Eq. (3.12) defined on the loop l(k j , kl ) = {(ki , k j , kl )| − π ≤ ki ≤ π } is nonzero [16, 17], where (ki , k j , kl ) are perpendicular to each other, and l(k j , kl ) does not intersect with the line node. For superconductors in class (ii), l(k j , kl ) can be defined for arbitrary surface direction since the operator T¯ corresponds to a pure TRS. Thus, an Majorana flat band appears on the superconductors’ surface in analogy with high-Tc cuprate superconductors [18, 19]. In class (iii) superconductors, on the other hand, T¯ corresponds to a magnetic translation, so l(k j , kl ) needs to be compatible with a translation vector t T , i.e., a Majorana flat band only arises when [t T ]i = 0, where [·]i means a vector component normal to the surface. Note that the behavior we have outlined here is similar to that of surface states in AFM topological insulators [20–27]. Thus, a

52

3 Superconducting Gap Classification on High-Symmetry Planes

limitation on possible Majorana flat bands in class (iii) superconductors appears in contrast to Majorana flat bands in class (ii) superconductors (and noncentrosymmetric superconductors [28–30]). In particular, when t T is perpendicular to the mirror plane, Majorana flat bands do not exist on any surface because no surface direction simultaneously satisfies the constraints arising from the magnetic translation and the mirror symmetry. Thus, a distortion or interaction that breaks the mirror symmetry is necessary to reveal the hidden Majorana flat bands.

3.3 Example: Sr2 IrO4 in − + +− State In this section, we suggest that superconductivity with nonsymmorphic-symmetryprotected gap structures in Table 3.1 appear in the − + +− magnetic state of Sr2 IrO4 , which is regarded as a higher-order MO order.

3.3.1 Background A layered perovskite 5d transition metal oxide Sr2 IrO4 has attracted much attention because a lot of similarities to the high-Tc cuprate superconductors have been recognized. For example, Sr2 IrO4 (La2 CuO4 ) has one hole per Ir (Cu) ion, and shows a pseudospin-1/2 AFM order [31]. Moreover, recent experiments on electron-doped Sr2 IrO4 indicate the emergence of a pseudogap [32–34] and at low temperatures a d-wave gap [35], which strengthens the analogy with cuprates. Furthermore, d-wave superconductivity in Sr2 IrO4 by carrier doping is theoretically predicted by several studies [36–39]. On the other hand, there are distinct differences of Sr2 IrO4 from cuprates: large SOC and nonsymmorphic crystal structure. Both of which attract interest in the modern condensed matter physics. Below TN 230 K, an AFM order develops in undoped Sr2 IrO4 . Large SOC and rotation of octahedra lead to canted magnetic moments from the a axis and induce a small FM moment along the b axis (Fig. 3.4). Several magnetic structures for stacking along the c axis have been reported in response to circumstances. The magnetic ground states determined by resonant x-ray scattering [40–42], neutron diffraction [43, 44], and second-harmonic generation [45], are summarized in a recent theoretical work [46]. In the undoped compound, the FM component shows the stacking pattern − + +− [40, 41, 43], as illustrated by the black arrows in Fig. 3.4. On the other hand, the + + ++ pattern is suggested as the magnetic structure of Sr2 IrO4 in a magnetic field directed in the ab plane [40] and of Rh-doped Sr2 Ir1−x Rhx O4 [42, 44]. The recent observation [45], however, advocates the − + −+ magnetic pattern indicating an intriguing odd-parity hidden order in Sr2 IrO4 (see the red arrows in Fig. 3.4).

3.3 Example: Sr2 IrO4 in − + +− State

(a) z = 1/8

53

(b) z = 3/8 FM “+” FM “+”

a−

b−

c−

d−

FM “−” FM “−”

(c) z = 5/8

(d) z = 7/8 FM “−” FM “+”

b+

a+

d+

c+

FM “+” FM “−” Fig. 3.4 Crystal and magnetic symmetries of Sr2 IrO4 in the 4 IrO2 planes: a z = 18 , b z = 38 , c z = 58 , and d z = 78 . The two magnetic patterns of interest, − + +− (black arrows) and − + −+ (red arrows), are shown. They differ by the FM in-plane component along the b axis. Iridium atoms (yellow circles) are labeled as a− , . . . , d− , a+ , . . . , d+ . Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society

The crystal space group of Sr2 IrO4 was originally reported as I 41 /acd according to neutron powder diffraction experiments [47, 48]. However, the crystal structure has recently been revealed by single-crystal neutron diffraction to be rather I 41 /a [44]. In either case, the symmetry of Sr2 IrO4 is globally centrosymmetric and nonsymmorphic. On the other hand, the site symmetry of the Ir site is S4 lacking local IS. In such noncentrosymmetric systems, ASOC entangles various internal degrees of freedom, such as spin, orbital, and sublattice. As an interesting consequence of the ASOC, locally noncentrosymmetric systems may have an advantage in realizing an odd-parity multipole order [49–56] beyond the paradigm of even-parity multipole order in d- and f -electron systems [57]. Based on the above background, we study unconventional superconducting state coexisting with a magnetic order in Sr2 IrO4 . In the following, we first reinterpret the − + +− and − + −+ orders in the context of magnetic multipole moments. Next, we show that superconductivity with nonsymmorphic symmetry-protected gap structures is induced by the − + +− order, which is regarded as a higher-order MO order.1 The result is evidenced by a combination of group-theoretical analysis and numerical analysis of an effective Jeff = 1/2 model for Sr2 IrO4 . this thesis, we do not discuss the FFLO superconductivity in the − + −+ state [10], because it digresses from the main subject.

1 In

54

3 Superconducting Gap Classification on High-Symmetry Planes

3.3.2 Classification of − + +− and − + −+ Orders Based on Magnetic Multipole Before going to the main issue, we show that the − + +− and − + −+ orders are classified into MO and MQ orders, respectively.

3.3.2.1

− + +− Order

Although the crystal point group of Sr2 IrO4 is D4h , it reduces to D2h in the − + +− ordered state. In Table 3.5, even-parity IRs of D4h except A1g , namely A2g , B1g , B2g , and E g , are subduced to representations of D2h . Since only B1g contains the fully symmetric IR of D2h (Ag ), the − + −+ order belongs to B1g representation of D4h . The lowest-order TRS-odd basis function of B1g is αx ysz + βz(ysx + xs y ) in the real space. This basis function represents an even-parity MO (l = 3) order [58],

Mˆ l,m

Mˆ 3,−2 − Mˆ 3,2 ∝ x y zˆ + yz xˆ + zx yˆ ,  n



2l j + 2s j · ∇ j r lj Z l,m (ˆr j )∗ , = μB l + 1 j=1

(3.18) (3.19)

 4π where Z l,m (ˆr ) ≡ 2l+1 Yl,m (ˆr ) is the normalized spherical harmonics. Thus, the − + +− order is classified into a MO order.

3.3.2.2

− + −+ Order

In the − + −+ ordered state, the crystal symmetry reduces from D4h to C2v . Here, the odd-parity IRs of D4h (A1u , A2u , B1u , B2u , and E u ) are subduced to representations of C2v (Table 3.6). Since only E u contains the fully symmetric IR of C2v (A1 ), the − + −+ order belongs to E u representation of D4h .

Table 3.5 Irreducible decomposition of D4h even-parity IRs in D2h point group. Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society (IRs of D4h ) A2g B1g B2g Eg (IRs of D4h )↓ D2h

B1g

Ag

B2g + B3g

B1g

Table 3.6 Irreducible decomposition of D4h odd-parity IRs in C2v point group. Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society (IRs of D4h ) A1u A2u B1u B2u Eu (IRs of D4h )↓ C2v

A2

B2

A2

B2

A1 + B1

3.3 Example: Sr2 IrO4 in − + +− State

55

This IR E u permits TRS-odd basis functions: αysz + βzs y in the real space, and k x in the momentum space. In the real space, the basis function contains an odd-parity MQ (l = 2) order [58], (3.20) Mˆ 2,1 + Mˆ 2,−1 ∝ y zˆ + z yˆ . Therefore, the − + −+ order contains the component of a MQ order, though it may include a toroidal dipole order proportional to y zˆ − z yˆ [49].

3.3.3 Superconducting Gap Classification Now we consider the superconductivity in the − + +− state. The magnetic space group of the − + +− state is a nonsymmorphic group PI cca = Pcca + {T | 2xˆ + yˆ + 2zˆ }Pcca. Since the group contains screw symmetry and magnetic translation 2 symmetry, nontrivial line nodes (gap opening) should appear on high-symmetry planes from the knowledge of Sects. 3.1 and 3.2. Thus, we classify gap structures on three BPs k z,x,y = 0 and three ZFs k z = π and k x,y = π . With the help of Tables 3.1 and 3.3, the representation P¯ k of the Cooper pair wave functions can be calculated. Then we induce the representation to the original crystal symmetry, P¯ k ↑ D4h , and decompose it into IRs of D4h . The obtained results are summarized in the following: • On the horizontal planes k z = 0 and π , P¯ k ↑ D4h

⎧ ⎪ ⎨ A1g + A2g + B1g + B2g BP, = + 2 A1u + 2 A2u + 2B1u + 2B2u + 2E u ⎪ ⎩2E + A + A + B + B + 4E ZF; g 1u 2u 1u 2u u

(3.21)

• On the vertical planes k x,y = 0 and π ,  ¯k

P ↑ D4h =

A1g + B1g + E g + 2 A1u + A2u + 2B1u + B2u + 3E u BP, ZF. A2g + B2g + E g + 3A1u + 3B1u + 3E u (3.22)

We find that possible IRs change from BPs to ZFs as a consequence of the nonsymmorphic symmetry. From Eqs. (3.21) and (3.22), for instance, gap structures for A1g and B2g superconducting states are found to have the nonsymmorphic feature as summarized in Table 3.7.

3.3.4 Numerical Calculation In this section, we demonstrate the results of group theory (Table 3.7) using an effective model for Jeff = 1/2 manifold.

56

3 Superconducting Gap Classification on High-Symmetry Planes

Table 3.7 The gap structure for A1g and B2g gap functions. Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society Order parameter k z = 0 kz = π k x,y = 0 k x,y = π A1g (s-wave) B2g (dx y -wave)

3.3.4.1

Gap Gap

Node Node

Gap Node

Node Gap

Model

Here a 3D single-orbital tight-binding model is introduced, which describes superconductivity coexisting with magnetic order in Sr2 IrO4 . The BdG Hamiltonian has a form of 1 † C (k) Hˆ BdG (k)C(k), (3.23) HBdG = 2 k

where † † † † C † (k) = (a−↑ (k+ ), a−↓ (k+ ), . . . , d−↑ (k+ ), d−↓ (k+ ), † † † † a+↑ (k+ ), a+↓ (k+ ), . . . , d+↑ (k+ ), d+↓ (k+ ),

a−↑ (k− ), a−↓ (k− ), . . . , d−↑ (k− ), d−↓ (k− ), a+↑ (k− ), a+↓ (k− ), . . . , d+↑ (k− ), d+↓ (k− )),

(3.24)

with k+ ≡ k + q2 , k− ≡ −k + q2 are 32-dimensional vector of creation-annihilation operators. The center-of-mass momentum q of Cooper pairs is assumed to be zero in † † (k), . . . d±s (k) are creation most cases except for the studies of FFLO state [10]. a±s operators of electrons with spin s =↑, ↓ on the sublattices a± , . . . , d± , respectively (for the sublattice symbols, see Fig. 3.4). The 32 × 32 BdG Hamiltonian matrix is ˆ described by a normal-state Hamiltonian Hˆ n (k) and an order parameter matrix (k), Hˆ BdG (k) = where



 ˆ Hˆ n (k+ ) (k) , † ˆ (k) − Hˆ nT (k− )

(3.25)

Hˆ n (k) = Hˆ kin (k) + Hˆ ASOC (k) + Hˆ M .

(3.26)

The kinetic term Hˆ kin (k) is given by ⎛

Hˆ intra-layer (k) Hˆ inter-layer2 (k) ⎜ ˆ + H (k) ⎜ inter-layer1 Hˆ kin (k) = ⎜ ⎝ ˆ Hˆ intra-layer (k) Hinter-layer2 (k)† + Hˆ inter-layer1 (k)

⎞ ⎟ ⎟ ⎟, ⎠

(3.27)

3.3 Example: Sr2 IrO4 in − + +− State

57

where (layer) (spin) (spin) {(ε2 (k) − μ)σ0(sl) σ0 + ε1 (k)σx(sl) σ0 }, Hˆ intra-layer (k) = σ0

Hˆ inter-layer1 (k) Hˆ inter-layer2 (k)

(spin) (spin) y = σx(layer) {Re(ε3x (k))σ0(sl) σ0 + Re(ε3 (k))σx(sl) σ0 } (spin) (spin) y − σ y(layer) {Im(ε3x (k))σ0(sl) σ0 + Im(ε3 (k))σx(sl) σ0 }, (spin) (spin) y = σx(layer) {Re(ε3 (k))σ0(sl) σ0 + Re(ε3x (k))σx(sl) σ0 } (spin) (spin) y + σ y(layer) {Im(ε3 (k))σ0(sl) σ0 + Im(ε3x (k))σx(sl) σ0 }, (spin)

(3.28)

(3.29)

(3.30)

(layer)

, σi(sl) , and σi are Pauli matrices representing with a chemical potential μ. σi spin, sublattice, and layer degrees of freedom, respectively. The kinetic energy terms x,y ε1 (k), ε2 (k), and ε3 (k) are written by ky kx cos , 2 2 ε2 (k) = −2t2 (cos k x + cos k y ), k x,y −ikz /4 x,y e , ε3 (k) = −t3 cos 2 ε1 (k) = −4t1 cos

(3.31) (3.32) (3.33)

which correspond to the nearest-, next-nearest-, and third-nearest-neighbor hoppings, respectively. Furthermore, the violation of local IS which induces the staggered ASOC, Hˆ ASOC (k), plays an essential role in an unconventional superconducting state in Sr2 IrO4 .2 This term is given by the following matrix: ⎛

⎞ Hˆ ASOC-intra1 (k) ⎜ + Hˆ x (k) ⎟ Hˆ ASOC-inter ASOC-intra2 (k) ⎜ ⎟ ⎜ ˆy ⎟ + H (k) ⎜ ⎟ ASOC-inter Hˆ ASOC (k) = ⎜ ⎟. ˆ ⎜ HASOC-intra1 (k) ⎟ ⎜ ⎟ x ⎝ Hˆ ASOC-inter (k)† + Hˆ ASOC-intra2 (k) ⎠ y + Hˆ ASOC-inter (k)

(3.34)

We take into account two intra-layer terms Hˆ ASOC-intra1 (k) and Hˆ ASOC-intra2 (k), and x,y two inter-layer terms Hˆ ASOC-inter (k):

ASOC is derived from atomic LS coupling [59, 60]. In our effective Jeff = 1/2 model, therefore, the ASOC takes account of the effect of LS-type SOC. Since multiorbital electronic structure does not play any essential role in our results, we will obtain qualitatively the same results in the 5d multiorbital model.

2 The

58

3 Superconducting Gap Classification on High-Symmetry Planes

k y (layer) (sl) (spin) kx cos σ0 Hˆ ASOC-intra1 (k) = iα1 cos iσ y σz , (3.35) 2 2 (layer) (sl) (spin) (spin) σz (sin k x cos k y σx − sin k y cos k x σ y ), (3.36) Hˆ ASOC-intra2 (k) = α2 σz 

 k k k k (layer) (sl) z x (spin) z x (spin) x cos Hˆ ASOC-inter (k) = −α3 iσ y iσ y − 2 sin sin σx cos σz 4 2 4 2

 kz k x (spin) k x (spin) kz (layer) (sl) sin , sin σx cos σz iσ y + 2 cos +iσx 4 2 4 2

(3.37)



 k y (spin) k y (spin) kz kz (layer) (sl) y cos sin σ y cos σz iσ y − 2 sin Hˆ ASOC-inter (k) = α3 iσ y 4 2 4 2

 k k kz kz y (spin) y (spin) (layer) (sl) sin , sin σ y cos σz iσ y + 2 cos −iσx 4 2 4 2

(3.38) which are allowed by the crystal symmetry of Sr2 IrO4 . The last term in Eq. (3.26), Hˆ M , expresses the molecular field of magnetic orders, − + +− and − + −+. As shown in Fig. 3.4, each site has the in-plane canted magnetic moment. Thus, the molecular field is given by ⎛ ⎜ Hˆ M = ⎝

−h(θa− ) · σ

⎞ ..

.

⎟ ⎠,

(3.39)

−h(θd+ ) · σ where (θa− , . . . , θd− , θa+ , . . . , θd+ )  (348◦ , 192◦ , 168◦ , 12◦ , 168◦ , 12◦ , 348◦ , 192◦ ) (− + +− state) = (3.40) (348◦ , 192◦ , 168◦ , 12◦ , 348◦ , 192◦ , 168◦ , 12◦ ) (− + −+ state), and h(θ ) = h(cos θ, sin θ, 0) [46]. ˆ Next we describe the order parameter (k). For on-site s-wave superconductivity, the matrix is given by ˆ (s) (k) = 0 1ˆ 2 σ0(layer) σ0(sl) iσ y(spin) .

(3.41)

On the other hand, when dx y -wave superconductivity arising from interaction between nearest-neighbor sites is assumed, the order parameter is ˆ (d) (k) = 0 sin

ky kx (layer) (sl) sin 1ˆ 2 σ0 σx iσ y(spin) . 2 2

(3.42)

Finally, we show the parameters used in the calculation. The hopping parameters of the effective Jeff = 1/2 model [37], which is derived from the three-orbital Hub-

3.3 Example: Sr2 IrO4 in − + +− State

59

Fig. 3.5 Contour plots of quasiparticle energy dispersion E/ 0 in the s-wave superconducting state, which is normalized by the magntiude 0 of the order parameter, on a k z = 0, b k z = π , c k x = 0, and d k x = π . The insets in a, b, c, and d show the dispersion E/ 0 along the respective blue line. Line nodes (black lines) appear on the ZFs k z = π and k x = π . Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society

bard model, are adopted: t1 = 1, t2 = 0.26, and t3 = 0.1. Furthermore, we assume moderate ASOC parameters α1 = 0.3 and α2 = α3 = 0.1, whose effects are visible in the numerical results. The chemical potential is set to be μ = 1.05, which results in the electron density n ∼ 1.2 where superconductivity has been predicted in a previous theoretical study [37]. Then, four spinful energy bands cross the Fermi level. The magnitude of gap function is chosen to be 0 = 0.02. Note that the numerical results shown below do not depend on the details of parameters, because they are eusured by the group-theoretical analysis.

60

3 Superconducting Gap Classification on High-Symmetry Planes

Fig. 3.6 Contour plots of normalized quasiparticle energy dispersion E/ 0 for the dx y -wave order parameter on a k z = 0, b k z = π , c k x = 0, and d k x = π . Line nodes (black lines) appear on the ZF k z = π and the BP k x = 0. Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society

3.3.4.2

Results

Now we show the numerical results of the model introduced above. By diagonalizing the BdG Hamiltonian [Eq. (3.25)], the quasiparticle energy dispersion in the superconducting state E = E(k x , k y , k z ) is obtained. The calculation results are shown in Figs. 3.5 and 3.6. In the figures, only 0 ≤ E/ 0 < 2 region is colored, and especially nodal (E ∼ 0) points are plotted by black. The gap structure of the two superconducting states obviously reproduces Table 3.7; for both s-wave and dx y -wave superconductivity, the numerical results are consistent with the group theory. In other words, the gap nodes in Figs. 3.5 and 3.6 on the ZFs are protected by nonsymmorphic space group symmetry. Note that

3.3 Example: Sr2 IrO4 in − + +− State

61

exceptional cases of the gap classification in Table 3.7 appear in some accidentally degenerate region [4]. For example, we see such unexpected gap structures on the k y = π plane (for details, see Appendix). As introduced previously, both theory [36–39] and experiment [35] suggest dx y wave superconductivity analogous to cuprates.3 In this case, a horizontal line node appears on the ZF (k z = π ) in contrast to the usual dx y -wave state. Moreover, the gap opening at the other ZFs (k x,y = π ) is also nontrivial because the usual dx y -wave order parameter vanishes not only at BPs but also at ZFs. Therefore, superconductivity in the MO (− + +−) state of Sr2 IrO4 possesses nontrivial gap structures protected by the nonsymmorphic space group symmetry.

Appendix: Other Remarks for Sr2 IrO4 Accidental Gap of A1g State at k y = π in − + +− State The group-theoretical gap classification in Eqs. (3.21) and (3.22) reveals that A1g superconductivity in the − + +− state possesses vertical line nodes on the ZF k y = π , although the A1g IR is allowed on the BP k y = 0. In our numerical calculation, however, a small gap appears in the excitation spectrum on the ZF as well as the BP (see Fig. 3.7). That is because band structures in a normal state are accidentally fourfold degenerate all over the ZF k y = π in our model. This fourfold degeneracy is not protected by symmetry except for on some high-symmetry lines (for details, see the next subsection). As mentioned in Sect. 2.2, the group-theoretical gap classification is applied only to an intra-band order parameter. In the presence of (nearly) fourfold degeneracy, however, an inter-band order parameter may affect the excitation gap [4]. Then, line nodes expected from the gap classification are lost. Indeed, recent theoretical study has suggested that such a gap opening changes the nodal line to nodal loops in UPt3 [4], where the inter-band gap appears only on high-symmetry lines in the ZF. In our tight-binding model, however, fourfold degeneracy accidentally emerges on the entire ZF k y = π , and therefore, we obtain the fully gapped structure here. We believe that the gap at k y = π is lifted by taking into account all possible SOCs allowed in the magnetic space group symmetry.

of the π/4 rotation around the principal (c) axis, the dx y -wave superconductivity corresponds to the dx 2 −y 2 -wave state in cuprates.

3 Because

62

3 Superconducting Gap Classification on High-Symmetry Planes

Fig. 3.7 Contour plots of normalized quasiparticle energy dispersion E/ 0 in s-wave superconductivity on a k y = 0 and b k y = π . a On the BP k y = 0, quasiparticle in almost entire region except the boundary k z = π are gapped, which is compatible with the group-theoretical gap classification (Table 3.7). b On the ZF k y = π , there is no line node contrary to the gap classification. Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society

Symmetry-Protected Dirac Line Nodes on BZ Boundary in − + +− State We here show the presence of fourfold degenerate normal band structures on the BZ boundary in − + +− state. Due to the symmetry, the fourfold degeneracy appears at U -R, R-T , T -Y , and Y -S lines in the first BZ (Fig. 3.8). Using the magnetic little group on each line, we prove the symmetry protection of the degeneracy in the following.

Fig. 3.8 The first BZ for primitive orthorhombic lattice. Fourfold degenerate band structures appear on the red lines. Reprinted with permission from Ref. [10]. Copyright © 2017 by the American Physical Society

kz Z

T

U

R Γ

X kx

Y S

ky

3.3 Example: Sr2 IrO4 in − + +− State

63

On the U -R line (k y = π and k z = π ), the magnetic little group is given by Mk = G k + TG k , G = {E|0}T + k

{Mz | 2xˆ

(3.43) +

zˆ }T 2

+

{Mx | 2zˆ }T

+

{C2y | 2xˆ }T,

(3.44)

where T = {T I | 2xˆ + 2yˆ + 2zˆ }. The fourfold degeneracy is proven from algebra,     ({C2y | 2xˆ })2 = −1, {C2y | 2xˆ }, {Mx | 2zˆ } = 0, and {C2y | 2xˆ }, T = 0 [4, 61, 62]. Because of the rotation symmetry {C2y | 2xˆ }, the normal part Hamiltonian on the U -R line is block diagonalized and decomposed into the ±i subsectors. The T symmetry is preserved in each subsector as ensured by the anticommutation relation between {C2y | 2xˆ } and T. Thus, Kramers pairs are formed in each subsector. The anticommutation relation between {C2y | 2xˆ } and {Mx | 2zˆ } ensures that a Kramers pair in the i subsector is degenerate with another Kramers pair in the −i subsector. Thus, the fourfold degeneracy is protected by symmetry. On the other lines, the fourfold degeneracy is proved in a similar way. On  the R-T and Y -S lines, we use the relations, ({C2x | 2zˆ })2 = −1, {C2x | 2zˆ }, {M y | 2xˆ } = 0, and   {C2x | 2zˆ }, T = 0. Finally on the T -Y line, the fourfold degeneracy is proved by the     relations, ({M y | 2xˆ })2 = −1, {M y | 2xˆ }, {C2z | 2xˆ + 2zˆ } = 0, and {M y | 2xˆ }, T = 0.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Norman MR (1995) Phys Rev B 52:15093 Micklitz T, Norman MR (2009) Phys Rev B 80:100506(R) Kobayashi S, Yanase Y, Sato M (2016) Phys Rev B 94:134512 Yanase Y (2016) Phys Rev B 94:174502 Nomoto T, Ikeda H (2016) Phys Rev Lett 117:217002 Micklitz T, Norman MR (2017) Phys Rev B 95:024508 Nomoto T, Ikeda H (2017) J Phys Soc Jpn 86(2):023703 Micklitz T, Norman MR (2017) Phys Rev Lett 118:207001 Sumita S, Yanase Y (2018) Phys Rev B 97:134512 Sumita S, Nomoto T, Yanase Y (2017) Phys Rev Lett 119:027001 Kobayashi S, Sumita S, Yanase Y, Sato M (2018) Phys Rev B 97:180504(R) Fujimoto S (2006) J Phys Soc Jpn 75(8):083704 Kobayashi S, Shiozaki K, Tanaka Y, Sato M (2014) Phys Rev B 90:024516 Zhao YX, Schnyder AP, Wang ZD (2016) Phys Rev Lett 116:156402 Bzdušek T, Sigrist M (2017) Phys Rev B 96:155105 Ryu S, Hatsugai Y (2002) Phys Rev Lett 89:077002 Sato M, Tanaka Y, Yada K, Yokoyama T (2011) Phys Rev B 83:224511 Tanaka Y, Mizuno Y, Yokoyama T, Yada K, Sato M (2010) Phys Rev Lett 105:097002 Tanaka Y, Sato M, Nagaosa N (2012) J Phys Soc Jpn 81(1):011013 Mong RSK, Essin AM, Moore JE (2010) Phys Rev B 81:245209 Turner AM, Zhang Y, Mong RSK, Vishwanath A (2012) Phys Rev B 85:165120 Essin AM, Gurarie V (2012) Phys Rev B 85:195116 Miyakoshi S, Ohta Y (2013) Phys Rev B 87:195133 Yoshida T, Peters R, Fujimoto S, Kawakami N (2013) Phys Rev B 87:085134 Baireuther P, Edge JM, Fulga IC, Beenakker CWJ, Tworzydło J (2014) Phys Rev B 89:035410

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3 Superconducting Gap Classification on High-Symmetry Planes

26. 27. 28. 29. 30. 31.

Zhang RX, Liu CX (2015) Phys Rev B 91:115317 Bègue F, Pujol P, Ramazashvili R (2017) Phys Lett A 381(14):1268 Brydon PMR, Schnyder AP, Timm C (2011) Phys Rev B 84:020501 Yada K, Sato M, Tanaka Y, Yokoyama T (2011) Phys Rev B 83:064505 Schnyder AP, Brydon PMR, Timm C (2012) Phys Rev B 85:024522 Kim BJ, Jin H, Moon SJ, Kim JY, Park BG, Leem CS, Yu J, Noh TW, Kim C, Oh SJ, Park JH, Durairaj V, Cao G, Rotenberg E (2008) Phys Rev Lett 101:076402 Kim YK, Krupin O, Denlinger JD, Bostwick A, Rotenberg E, Zhao Q, Mitchell JF, Allen JW, Kim BJ (2014) Science 345(6193):187 Yan YJ, Ren MQ, Xu HC, Xie BP, Tao R, Choi HY, Lee N, Choi YJ, Zhang T, Feng DL (2015) Phys Rev X 5:041018 Battisti I, Bastiaans KM, Fedoseev V, Torre Adl, Iliopoulos N, Tamai A, Hunter EC, Perry RS, Zaanen J, Baumberger F, Allan MP (2017) Nat Phys 13:21 Kim YK, Sung NH, Denlinger JD, Kim BJ (2016) Nat Phys 12:37 Wang F, Senthil T (2011) Phys Rev Lett 106:136402 Watanabe H, Shirakawa T, Yunoki S (2013) Phys Rev Lett 110:027002 Yang Y, Wang WS, Liu JG, Chen H, Dai JH, Wang QH (2014) Phys Rev B 89:094518 Meng ZY, Kim YB, Kee HY (2014) Phys Rev Lett 113:177003 Kim BJ, Ohsumi H, Komesu T, Sakai S, Morita T, Takagi H, Arima T (2009) Science 323(5919):1329 Boseggia S, Walker HC, Vale J, Springell R, Feng Z, Perry RS, Sala MM, Rønnow HM, Collins SP, McMorrow DF (2013) J Phys Condens Matter 25(42):422202 Clancy JP, Lupascu A, Gretarsson H, Islam Z, Hu YF, Casa D, Nelson CS, LaMarra SC, Cao G, Kim YJ (2014) Phys Rev B 89:054409 Dhital C, Hogan T, Yamani Z, de la Cruz C, Chen X, Khadka S, Ren Z, Wilson SD (2013) Phys Rev B 87:144405 Ye F, Wang X, Hoffmann C, Wang J, Chi S, Matsuda M, Chakoumakos BC, Fernandez-Baca JA, Cao G (2015) Phys Rev B 92:201112 Zhao L, Torchinsky DH, Chu H, Ivanov V, Lifshitz R, Flint R, Qi T, Cao G, Hsieh D (2016) Nat Phys 12:32 Di Matteo S, Norman MR (2016) Phys Rev B 94:075148 Huang Q, Soubeyroux J, Chmaissem O, Sora IN, Santoro A, Cava RJ, Krajewski JJ, Peck WF Jr (1994) J Solid State Chem 112:355 Crawford MK, Subramanian MA, Harlow RL, Fernandez-Baca JA, Wang ZR, Johnston DC (1994) Phys Rev B 49:9198 Spaldin NA, Fiebig M, Mostovoy M (2008) J Phys Condens Matter 20(43):434203 Yanase Y (2014) J Phys Soc Jpn 83(1):014703 Hitomi T, Yanase Y (2014) J Phys Soc Jpn 83(11):114704 Hayami S, Kusunose H, Motome Y (2014) Phys Rev B 90:024432 Hayami S, Kusunose H, Motome Y (2014) Phys Rev B 90:081115 Hayami S, Kusunose H, Motome Y (2015) J Phys Soc Jpn 84(6):064717 Fu L (2015) Phys Rev Lett 115:026401 Hitomi T, Yanase Y (2016) J Phys Soc Jpn 85(12):124702 Kuramoto Y, Kusunose H, Kiss A (2009) J Phys Soc Jpn 78(7):072001 Schwartz C (1955) Phys Rev 97:380 Zhong Z, Tóth A, Held K (2013) Phys Rev B 87:161102 Yanase Y (2013) J Phys Soc Jpn 82(4):044711 Liang QF, Zhou J, Yu R, Wang Z, Weng H (2016) Phys Rev B 93:085427 Sumita S, Yanase Y (2016) Phys Rev B 93:224507

32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

Chapter 4

Superconducting Gap Classification on High-Symmetry Lines

4.1 Group-Theoretical Classification The magnetic little group on an n-fold axis Mnk is given by Mnk = Gnk + TGnk =

n−1  m=0

{Cn |0}m T + T

n−1 

{Cn |0}m T,

(4.1)

m=0

where Cn represents the n-fold rotation, and T = {T I |0} in PM superconductors.1 The small corepresentations of Mnk are obtained by the double-valued IR of corresponding point groups (little cogroups) Cn ∼ = Gnk /T (Table 4.1a–d). Note that each IR in Table 4.1 is composed of two 1D representations, which are degenerate due to the pseudo-TRS T. The subscripts 1/2, 3/2, and 5/2 correspond to total angular momentum of the normal Bloch state jz = ±1/2, ±3/2, and ±5/2, respectively [2, 3]. In some other cases, more generally, a high-symmetry axis in the BZ has a vertical mirror symmetry where the mirror plane contains the axis. Therefore, the general ¯ k = Cnv + TCnv ¯ k = Cn + TCn (without mirrors) or M magnetic little cogroup is M (with mirrors). From the small corepresentation corresponding to the normal Bloch wave function, the representation P¯αk of the Cooper pair wave function is calculated using the Mackey-Bradley theorem (Theorem 2.1). The obtained results of P¯αk are summarized in Table 4.2. According to Table 4.2, the pair wave function on C2(v) - and C4(v) -symmetric axes has a unique representation irrespective of the IR α. On C3(v) - and C6(v) -symmetric lines, on the other hand, we find two nonequivalent representations P¯αk of the pair wave function depending on α, namely the total angular momentum jz of the normal Bloch state. For example, the E u order parameter on a threefold line is allowed (forbidden) when the Bloch state belongs to the E 1/2 (2B3/2 ) representation. This representatives in Mnk /T can be chosen to have no translation operation since we assume symmorphic space groups. This does not mean the unimportance of nonsymmorphic symmetry on high-symmetry lines. Indeed, recent theoretical study has pointed out that the gap classification on C2v -symmetric hinges of the BZ with glide symmetry is different from that with mirror symmetry [1]. 1 All

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Sumita, Modern Classification Theory of Superconducting Gap Nodes, Springer Theses, https://doi.org/10.1007/978-981-33-4264-4_4

65

66

4 Superconducting Gap Classification on High-Symmetry Lines

Table 4.1 Double-valued IRs of cyclic point groups: (a) C2 , (b) C3 , (c) C4 , and (d) C6 [2–5]. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society (a) Twofold axis (b) Threefold axis C2

E C2

E 1/2

2 0

(c) Fourfold axis

C3

E C3 C32

E 1/2

2 1

2B3/2

2 −2 2

−1

C4 E 1/2 E 3/2

E C4 C43 C42 √ √ 2 2 − 2 0 √ √ 2 − 2 2 0

(d) Sixfold axis

E 5/2

E C6 C65 C3 C32 C2 √ √ 2 3 − 3 1 −1 0 √ √ 2 − 3 3 1 −1 0

E 3/2

2 0

C6 E 1/2

0

−2 2

0

Table 4.2 Group-theoretical superconducting gap classification on high-symmetry k lines where the little cogroup is denoted by G¯ k [5, 6]. The representation P¯αk of the Cooper pair wave function k = G¯ k + I G¯ k . The labels of IR on each line is decomposed by IRs of the corresponding group G¯pair (α) are represented by the subscripts 1/2, 3/2, and 5/2. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society k ) P¯αk λ¯ αk G¯ k (G¯pair C2 (C2h ) C3 (S6 ) C4 (C4h ) C6 (C6h ) C2v (D2h ) C3v (D3d ) C4v (D4h ) C6v (D6h )

E 1/2 E 1/2 2B3/2 E 1/2 , E 3/2 E 1/2 , E 5/2 E 3/2 E 1/2 E 1/2 E 3/2 E 1/2 , E 3/2 E 1/2 , E 5/2 E 3/2

A g + Au + 2Bu A g + Au + E u A g + 3Au A g + Au + E u A g + Au + E 1u A g + Au + 2Bu A g + Au + B2u + B3u A1g + A1u + E u A1g + 2 A1u + A2u A1g + A1u + E u A1g + A1u + E 1u A1g + A1u + B1u + B2u

means that the E u superconducting gap opens for an energy band with jz = ±1/2, while nodes appear for a jz = ±3/2 band. Therefore, the presence or absence of nodes is jz -dependent when the system has threefold or sixfold rotational symmetry. In this thesis, we call it a jz -dependent gap structure, which is not obtained by the Sigrist–Ueda method (see Sect. 1.1.2.3).

4.2 Topological Classification Here we classify gap structures on high-symmetry k lines in terms of topology, using the topological classification based on the EAZ symmetry class (Sect. 2.3). The classification results are summarized in Table 4.3. Note that the EAZ symmetry classes in the table are in general different from the AZ classes of the total BdG Hamiltonian; the EAZ classes represent the symmetry of the block-diagonalized Hamiltonian instead, which is labeled by the IRs of G¯ k (see also the example in Sect. 2.4). When

2 E [1 E ] D g g 1 E [2 E ] A g g 2 E [1 E ] C u u 1 E [2 E ] A u u (d1) G¯ k = C6 , α = +[−]1/2, +[−]5/2

Z2 (L)

0 (G)

D

C

Bg

Bu

Au Bg Bu 1E

0 (G)

Z (L)

Z (P)

Z2 (S)

Z (S)

0 (G)

Z (P)

AIII

A

A

D

A

C

A

Au

Bg

Bu 2 E [1 E ] g g 1 E [2 E ] g g 2 E [1 E ] u u 1 E [2 E ] u u 2 1g [ E 1g ] 2 E [1 E ] 1g 1g 1 E [2 E ] 1u 1u 2 E [1 E ] 1u 1u 1,2 E 2g 1,2 E 2u

Ag

AIII

Ag

IR of C6h

Classification

0 (G)

EAZ

IR of C4h

(c) G¯ k = C4 , α = +[−]1/2, +[−]3/2

Au

0 (G)

AIII

u

g

A

A

CII

DIII

2g 2u

1,2 E 1,2 E

Z (S) Z (P)

A

1u

1g

A

1,2 E

Bu 1,2 E

Bg

Au

Ag

IR of C6h

A

A

A

A

C

D

AIII

AIII

EAZ

(d2) G¯ k = C6 , α = ±3/2

1,2 E

1,2 E

Au

Ag

EAZ

(b2) G¯ k = C3 , α = ±3/2 IR of S6

Z (P)

0 (G)

Z (S)

Z2 (S)

Z (P)

Z (L)

0 (G)

0 (G)

Classification

Z (P)

0 (G)

Z (S)

Z2 (S)

0 (G)

0 (G)

Classification

A

C

A

D

A

A

AIII

AIII

EAZ

AIII

AIII

Au

Ag

0 (G)

AIII

Ag

EAZ

(b1) G¯ k = C3 , α = +[−]1/2 IR of S6

Classification

EAZ

IR of C2h

(a) G¯ k = C2 , α = ±1/2

Z (P)

Z (S)

Z (P)

Z (S)

0 (G)

(continued)

Z2 (L)

0 (G)

0 (G)

Classification

Z (P)

Z (S)

0 (G)

0 (G)

Classification

Table 4.3 Topological gap classification on high-symmetry lines in the BZ. Each classification is characterized by the type of the topological number and the gap structure: (G) full gap, (P) point nodes, (L) line nodes, and (S) surface nodes (Bogoliubov FSs). In a spontaneously TRS breaking phase, 2D IRs in D3d , D4h , and D6h are the same as those in S6 , C4h , and C6h , respectively, since all the 2D IRs are decomposed into 1D IRs with different eigenvalues of the rotation symmetry (see also Table 4.4). Therefore we do not show the 2D IRs in the tables (f), (g), and (h). Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

4.2 Topological Classification 67

A2u

2D IRs

Z2 (L)

Z2 (P)

Z2 (L)

0 (G)

Z2 (L)

0 (G)

BDI

BDI

BDI

CI

BDI

CI

B1g

B1u

B2g

B2u

B3g

B3u

see (b1)

BDI

BDI

CI

CI

A1u A2g A2u B1g B1u B2g B2u

0 (G)

Z2 (L)

Z2 (P)

Z (L)

Z (P)

Z (L)

Z (P)

CI

BDI

BDI

AI

AI

AI

AI

See (c)

A1u

A2g

A2u

B1g

B1u

B2g

B2u

2D IRs 2D IRs

A1g

0 (G)

CI

A1g

IR of D6h

Classification

EAZ

IR of D4h

See (d1)

AI

AI

AI

AI

BDI

BDI

CI

CI

EAZ

(h1) G¯ k = C6v , α = 1/2, 5/2

A2g

0 (G)

CI

Au

(g) G¯ k = C4v , α = 1/2, 3/2

A1u

0 (G)

(f1) G¯ k = C3v , α = 1/2 EAZ

IR of D3d A1g

CI

Classification

IR of D2h Ag

(e) G¯ k = C2v , α = 1/2 EAZ

Table 4.3 (continued)

Z (P)

Z (L)

Z (P)

Z (L)

Z2 (P)

Z2 (L)

0 (G)

0 (G)

Classification

Z2 (P)

Z2 (L)

0 (G)

0 (G)

Classification

see (b2)

AIII

D

C

2D IRs

B2u

B2g

B1u

B1g

A2u

A2g

A1u

A1g

IR of D6h

See (d2)

CI

BDI

CI

BDI

BDI

BDI

CI

CI

EAZ

(h2) G¯ k = C6v , α = 3/2

2D IRs

A2u

A2g

A1u

AIII

(f2) G¯ k = C3v , α = 3/2 EAZ

IR of D3d A1g

0 (G)

Z2 (L)

0 (G)

Z2 (L)

Z2 (P)

Z2 (L)

0 (G)

0 (G)

Classification

0 (G)

Z2 (L)

0 (G)

0 (G)

Classification

68 4 Superconducting Gap Classification on High-Symmetry Lines

4.2 Topological Classification

69

Table 4.4 Character tables for 2D IRs of the point groups (a) S6 , (b) C4h , and (c) C6h . All notations are based on Bilbao Crystallographic Server [7]. It is noteworthy that all representations labeled by the indices 1 and 2 are 1D IRs, which are doubly degenerated under TRS (e.g., E u = 1 E u + 2 E u ). In each table, characters are shown only for generators of the corresponding group. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society (a) S6 E C3 I (c) C6h E C6 I (3+ ) + g (2 ) 1 E ( − ) u 2 2 E ( − ) u 3 1E

g

2E

(b) C4h 1 E ( + ) g 4 2 E ( + ) g 3 1 E ( − ) u 4 2 E ( − ) u 3

1 1 1 1 E 1 1 1 1

e−i2π/3 e+i2π/3 e−i2π/3 e+i2π/3 C4 −i +i −i +i

+1 +1 −1 −1 I +1 +1 −1 −1

1E

1g

2E

1g

1E

1u

2E

1u

1E

2g

2E

2g

1E

2u

2E

2u

(6+ ) (5+ ) (6− ) (5− ) (3+ ) (2+ ) (3− ) (2− )

1 1 1 1 1 1 1 1

e−iπ/3 e+iπ/3 e−iπ/3 e+iπ/3 −e−iπ/3 −e+iπ/3 −e−iπ/3 −e+iπ/3

+1 +1 −1 −1 +1 +1 −1 −1

the IR of the normal Bloch state and that of the superconducting order parameter are given, it is identified, from Table 4.3, whether the superconducting gap closes or not on almost all high-symmetric axes with some exceptions.2 ! Remark Now we remark the treatment of the 2D IRs in Table 4.3. A 2D superconducting order ˆ + (k) + η−  ˆ − (k). For examˆ parameter generally possesses the matrix form (k) = η+  ple, we consider an order parameter belonging to the E g IR of C3 , which satisfies ˆ ± (k), ˆ ± (k)u −k (C3 )T = e±i2π/3  u k (C3 )

(4.2)

on the C3 -symmetric line. Here u T means the transpose of the matrix u. In general, it is impossible to determine a commutation relation between the rotation symmetry C3 and the PHS C for arbitrary parameters η+ and η− . However, if we fix the order parameter ˆ ˆ ± (k), namely (η+ , η− ) = (1, 0) or to one of the two rotation-invariant bases (k) = (0, 1), the commutation (factor system) is given by ωin (C3 , C) = e±i2π/3 ωin (C, C3 ).

(4.3)

ˆ ˆ + (k) ( ˆ − (k)) belongs to the 1D IR 2 E g (1 E g ) of S6 (see In other words, (k) = Table 4.4a). ˆ − (k) are a pair of the order parameter connected by ˆ + (k) and  Note that since  TRS, choosing one of them leads to TRS and CS breaking. In this case, therefore, we have only to calculate the Wigner criterion for the pseudo-PHS operator C: 2 For

example, see Ref. [1].

70

4 Superconducting Gap Classification on High-Symmetry Lines

1 C W+1/2 = 3 C W+3/2 =

1 3

 2 g∈{E,C ¯ 3 ,(C 3 ) }



g∈{E,C ¯ 3 ,(C 3

)2 }

k ω k (Cg, ¯ Cg)χ ¯ [γ¯+1/2 ((Cg)2 )] =

 0 1 Eg , 1 2 Eg ,

k ω k (Cg, ¯ Cg)χ ¯ [γ¯+3/2 ((Cg)2 )] = 0 (1,2 E g ),

(4.4) (4.5)

where the 1D IRs 1,2 E g of S6 are defined in Table 4.4a. For calculations in Eqs. (4.4) and (4.5), we use Eq. (4.3), ωin (C, C) = +1, and ωin (C3 , C3 )ωin ((C3 )2 , C3 ) = −1. The IRs k k and γ¯+3/2 of C3 are given in the following character table. γ¯+1/2 IR E C3 (C3 )2 k +iπ/3 γ¯+1/2 1 e e+i2π/3 k γ¯+3/2 1 −1 1

(4.6)

Equations (4.4) and (4.5) indicate that the classification results of 1 E g and 2 E g are different (equivalent) for the IR α = +1/2 (+3/2) of the normal Bloch state; see also Table 4.3b1 and b2. Furthermore, the classification results in Eq. (4.4) are reversed for an α = −1/2 band:  1 1 Eg , C (4.7) W−1/2 = 0 2 Eg , which are represented inside the square brackets in Table 4.3b1. So far we have considered the treatment of the 2D IR E g of C3 symmetry. In a similar way, all 2D IRs can be decomposed into the 1D IRs with different eigenvalues of the rotation symmetry (see Table 4.4). For such 1D IRs produced from 2D IRs, therefore, the groups D3d , D4h , and D6h are reduced to S6 , C4h , and C6h , respectively. Thus we do not show the 2D IRs for D3d , D4h , and D6h groups in Table 4.3.

In Table 4.3, the classification “0” means a fully gapped structure on the highsymmetry axis, which is consistent with the results of Table 4.2. When the classification is nontrivial (Z or Z2 ), on the other hand, the gap closes on the line. Taking the whole FS into account, such a gapless spectrum has three types: (P) point nodes, (L) line nodes, and (S) surface nodes (Bogoliubov FSs), which are shown in Fig. 4.1 The condition for each node structure is specified by the spatial parity and TRS of the order parameter as follows. Case (S): even-parity and TRS breaking order parameter. When both particle bands and hole bands are doubly degenerate, a gapless point on the high-symmetry axis are a point node, which are protected by a Chern number defined on a sphere wrapping around the node. Indeed, most single-band models with even-parity and TRS breaking order parameter reproduce the situation. In real TRS breaking superconductors where multi-band effects cannot be neglected, however, the degeneracy splits due to the inter-band pairing effect [8, 9]. As a result, the point nodes are inflated to surface

4.2 Topological Classification

(P)

71

(L)

(S)

FS

Fig. 4.1 Three cases of gapless structures on a high-symmetry axis: P point nodes, L a part of line nodes, and S a part of surface nodes (Bogoliubov FSs). Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

nodes, which are characterized by a 0D topological number (Pfaffian of the antisymmetrized total BdG Hamiltonian) P(k) ∈ Z2 [8]. The 0D topological invariant is defined only in even-parity superconductors. Case (L): even-parity and TRS preserving order parameter. When there exists a vertical mirror-invariant plane containing the high-symmetry axis (i.e., G¯ k ∼ = Cnv ), the presence or absence of line nodes on the mirror plane is already classified in Tables 3.1 and 3.4. Taking into account the compatibility relation between the mirror plane and the high-symmetry axis, it is found that nodes on the Cnv -symmetric axis are a part of line nodes on the plane. The line nodes are also protected by the 1D topological number (winding number) Wl ∈ Z in Eq. (3.12), which is defined in even-parity superconductors by using the CS  [10–12]. This indicates that the line nodes do not vanish as long as the CS is preserved even when the mirror symmetry is broken. In the case, the line nodes remain intersectant with the axis because of the rotational symmetry. Therefore, nodes on the Cn -symmetric axis are also a part of the line nodes. Case (P): odd-parity order parameter. Except for the BZ boundary in nonsymmorphic space groups, there is no line node in odd-parity superconductivity because of Blount’s theorem [11, 13] (see also Sect. 1.1.2.2). (P1) When the order parameter breaks TRS, band degeneracy in general splits due to the multi-band effect [8, 9]. However, nodes on the high-symmetry axis are not a part of surface nodes since no 0D topological number is defined at a general point in the 3D BZ. Thus the stable nodes are point nodes. (P2) When the order parameter preserves TRS, nodes on the high-symmetry axis are point nodes because the band splitting does not occur. In the following sections, we see some representative candidates of symmetryprotected nodes on high-symmetry axes.

72

4 Superconducting Gap Classification on High-Symmetry Lines

H

C

B

A

T

Fig. 4.2 Schematic magnetic field (H )–temperature (T ) phase diagram of UPt3 [5, 6, 20, 21, 25, 26]. Experimental studies have observed multiple superconducting phases: TRS preserving A and C phases, and TRS breaking B phase. The shaded region shows the Weyl superconducting phase [25, 26]. In the system, C3 symmetry is preserved only on the dashed line. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

4.3 UPt3 (Space Group: P63 /mmc) 4.3.1 Background Superconductivity in UPt3 has been intensively investigated after the discovery of superconductivity in 1980s [14]. Multiple superconducting phases illustrated in Fig. 4.2 [15–18] unambiguously show exotic Cooper pairing belonging to a 2D IR of point group D6h [19]. After several theoretical proposals examined by experiments for more than three decades, the E 2u representation has been considered to be most reasonable as symmetry of the superconducting order parameter [20, 21]. Especially, the multiple superconducting phases in the H –T plane are naturally reproduced by assuming a weak symmetry-breaking term of hexagonal symmetry [20]. Furthermore, a phase-sensitive measurement [22] and an observation of spontaneous TRS breaking [23, 24] in the low-temperature and low-magnetic-field B phase, which was predicted in the E 2u state, support the E 2u symmetry of superconductivity. Figure 4.3 shows the crystal structure of UPt3 , which is composed of two uranium sublattices. The crystal symmetry is characterized by a nonsymmorphic space group P63 /mmc.3 This space group is based on the primitive hexagonal Bravais lattice, the BZ of which is illustrated in Fig. 4.4. The BZ has two threefold rotation lines K -H and K  -H  , as well as a sixfold rotation axis -A. Quantum oscillation measurements combined with ab initio calculations [21, 28– 32] have shown a pair of FSs centered at the A point (A-FSs), three FSs at the  point (-FSs), and two FSs at the K point (K -FSs) in UPt3 . Superconducting gap structures on the K -FSs have not been theoretically studied, while the existence 3 Symmetry

breaking by a weak crystal distortion has been reported [27], although its reliability is under debate. We here assume the high symmetry space group P63 /mmc.

4.3 UPt3 (Space Group: P63 /mmc)

73

Fig. 4.3 Crystal structure of UPt3 . The filled and open circles show uranium ions forming AB stacked triangular lattice. The red and green arrows show the 2D vectors ei and r i (i = 1, 2, 3), respectively. The black solid diamond shows the unit cell. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society Fig. 4.4 The first BZ of primitive hexagonal Bravais lattice. The red lines show threefold and sixfold rotation symmetric axes. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society

A Γ

H H’

K

K’

of symmetry-protected line nodes and Weyl nodes on the A-FSs and -FSs have been suggested in previous theories [25, 26, 32–37]. From the classification results given in Tables 4.2 and 4.3, however, it would be interesting to examine the gap structures on the K -FSs, since they cross the threefold K -H line. Indeed, we show that jz -dependent point nodes appear on the K -H line.

4.3.2 Gap Classification on K -H Line Now we apply the gap classification in Tables 4.2 and 4.3 to the space group of UPt3 . As mentioned above, the hexagonal BZ possesses a sixfold axis -A, and threefold axes K -H and K  -H  (see Fig. 4.4). On the -A line, the magnetic little cogroup is ¯ k = C6v + TC6v symmetry which results in three IRs E 1/2 , E 3/2 , and E 5/2 , of the M normal Bloch state. Thus, Table 4.2 shows that two distinct representations of the Cooper pair wave function,  P¯ k =

A1g + A1u + E 1u (λ¯ αk = E 1/2 , E 5/2 ), A1g + A1u + B1u + B2u (λ¯ αk = E 3/2 ),

(4.8)

74

4 Superconducting Gap Classification on High-Symmetry Lines

are obtained in the superconducting state. Equation (4.8) is represented by IRs of point group D6h = C6v + I C6v . The same result was suggested by Yarzhemsky [33, 35]. In the above background, we explained that UPt3 is considered to possess the E 2u order parameter. According to Eq. (4.8), the E 2u IR is prohibited for any representations E 1/2 , E 3/2 , and E 5/2 . Therefore, point nodes emerge on the -A line irrespective of the character of the normal Bloch state. The same conclusion is derived from the Sigrist–Ueda method. On the other hand, the gap structure on the K -H (K  -H  ) line is jz -dependent. ¯ k = C3v + TC3v , whose IRs are λ¯ k = E 1/2 and The magnetic little cogroup is M E 3/2 . Corresponding to these two Bloch states, the group-theoretical classification (Table 4.2) of the Cooper pair wave function results in two nonequivalent representations:  A1g + A1u + E u (λ¯ αk = E 1/2 ), P¯ k = (4.9) A1g + 2 A1u + A2u (λ¯ αk = E 3/2 ), where we use irreducible decomposition in point group D3d = C3v + I C3v . Then, P¯ k can be induced to the original crystal point group D6h with the help of the Frobenius reciprocity theorem [4]. The induced representations P¯ k ↑ D6h are given by  ¯k

P ↑ D6h =

A1g + B2g + A1u + B2u + E 1u + E 2u (λ¯ αk = E 1/2 ), A1g + B2g + 2 A1u + A2u + B1u + 2B2u (λ¯ αk = E 3/2 ).

(4.10)

Equation (4.10) indicates that the E 2u superconducting gap opens for the E 1/2 normal Bloch state, while point nodes appear for E 3/2 . Thus, the gap structure is indeed jz dependent. In this case, the Sigrist–Ueda classification method breaks down since it has taken into account only the pseudospin degree of freedom s = 1/2. In terms of topology, furthermore, Table 4.3b1 and b2 also shows that the gap classification for the 1,2 E u order parameter depend on the angular momentum: 0 ⊕ Z for α = ±1/2,

(4.11a)

Z ⊕ Z for α = ±3/2.

(4.11b)

However, in this case, the topological classification is slightly different from the group-theoretical gap classification [Eq. (4.10)]. Equation (4.11) indicates that, for the α = ±1/2 states, one Bogoliubov quasiparticle spectrum is gapped (0), while the other is gapless (Z). Which one is gapped depends on the C3 eigenvalue of the order parameter (see also the remark in Sect. 4.2). The inconsistency is due to the fact that the spontaneous TRS breaking is taken into account not in the group-theoretical analysis, but in the topological argument. In the spontaneous symmetry breaking case the topological classification predicts a correct gap structure. In the following, we demonstrate the nontrivial jz -dependent gap structure by analyzing a microscopic model.

4.3 UPt3 (Space Group: P63 /mmc)

75

4.3.3 Model and Normal Bloch State Here we introduce a microscopic model of UPt3 , and clarify the property of band structures on the K -H line. The BdG Hamiltonian for a 3D two-sublattice model [25, 26] is given by HBdG

  ˆ 1 † (k) Hˆ n (k) C(k), = C (k) † ˆ (k) − Hˆ n (−k)T 2

(4.12)

k

with † † † † C † (k) = (ca↑ (k), ca↓ (k), cb↑ (k), cb↓ (k), ca↑ (−k), ca↓ (−k), cb↑ (−k), cb↓ (−k)), (4.13) where k, m (= a, b), and s (= ↑, ↓) are indices of momentum, sublattice, and spin, respectively. The BdG Hamiltonian matrix is composed of the normal-state Hamiltonian,   a(k)s0 ξ(k)s0 + α g(k) · s , (4.14) Hˆ n (k) = a(k)∗ s0 ξ(k)s0 − α g(k) · s

ˆ and the order-parameter matrix (k) = [(k)]ms,m  s  . Here si represents a Pauli matrix in spin space. Taking into account the crystal structure of UPt3 illustrated in Fig. 4.3, we adopt an intra-sublattice kinetic energy, ξ(k) = 2t



cos k · ei + 2tz cos k z − μ,

(4.15)

i=1,2,3

and an inter-sublattice hopping term, a(k) = 2t  cos

k z  i k ·r i e , 2 i=1,2,3

(4.16)

with k = (k , k y ). The basis translation vectors in two dimensions are e1 = (1, 0), √x √ e2 = (− 21 , 23 ), and e3 = (− 21 , − 23 ). The inter-layer neighboring vectors projected onto the x-y plane are given by r 1 = ( 21 , 2√1 3 ), r 2 = (− 21 , 2√1 3 ), and r 3 = (0, − √13 ). These 2D vectors are shown in Fig. 4.3. Although the global crystal point group D6h is centrosymmetric, the local point group symmetry at uranium ions is D3h lacking IS. Thus, the local symmetry allows a Kane–Mele type ASOC [38] with a g vector [39], g(k) = zˆ

 i=1,2,3

sin k · ei .

(4.17)

76

4 Superconducting Gap Classification on High-Symmetry Lines

The coupling constant is staggered between the two sublattices [(+α, −α) in Eq. (4.14)], so as to preserve the global D6h point group symmetry [38, 40, 41]. Before going to the analysis of superconducting gap structures, it is necessary to identify the property of the normal Bloch state. First, we calculate the normal energy bands on the K -H line from the normal part Hamiltonian [Eq. (4.14)]. Although the band structure is originally fourfold degenerated due to the two sublattices and two spin degrees of freedom, this splits into twofold + twofold degenerate bands as a result of the ASOC term. The schematic band structures are illustrated in Fig. 4.5a and b. When the coupling constant α of the ASOC is positive,4 |a, ↑ and |b, ↓ (|b, ↑ and |a, ↓) states cross the Fermi level on the K -H (K  -H  ) line (Fig. 4.5a). On the other hand, the spin states on the Fermi level are exchanged when α < 0, which is shown in Fig. 4.5b. Note that the bases of the energy bands are constructed by pure sublatticebased representations, since the inter-sublattice hopping term [Eq. (4.16)] vanishes on the K -H and K  -H  lines. This vanishing has been proved by the symmetry analysis [42, 43]. In order to apply the gap classification theory, next we should consider the following question: “Which representation does the band crossing the Fermi level belong to, E 1/2 ( jz = ±1/2) or E 3/2 ( jz = ±3/2)?” Let us investigate the problem in the next subsection.

4.3.4 Effective Orbital Angular Momentum In this subsection, we consider the total angular momentum jz of the normal Bloch states depicted in Fig. 4.5. Recalling the quantum mechanics, jz should be composed of an orbital angular momentum l z and a spin angular momentum sz . The former l z is equal to zero because the orbital degree of freedom is neglected in our two-sublattice single-orbital model [Eq. (4.14)]. Since electrons are spin-1/2 fermions, the latter spin angular momentum is sz = ±1/2. Therefore, the total angular momentum jz is apparently identified as l z + sz = ±1/2 in this model. However, this is not right, as we show below. In order to correctly calculate jz , we have to take into account an effective orbital angular momentum λz , which arises from the sublattice degree of freedom. The normal Bloch state has a phase factor (plane-wave part) ei k·r , which depends on the real-space coordinate r of the site [42, 43]. Figure 4.6 shows the phase factor for the normal Bloch state at the K point [k = (4π/3, 0, 0)]. By operating threefold rotation on the K -point normal Bloch state, the a sublattice acquires the phase value e+i2π/3 (red arrows), while the b sublattice obtains e−i2π/3 (blue arrows). These phase factors correspond to eiλz θ (θ = 2π/3), and therefore, the sublattices a and b possess effective orbital angular momenta λz = +1 and −1, respectively. Note that the normal Bloch state at the K  point has a complex conjugate phase factor to that 4 Here, we dub the ASOC coupling constant as α according to the convention. Although the symbol is the same as that of the normal Bloch state, the difference is obvious from the context.

4.3 UPt3 (Space Group: P63 /mmc) Fig. 4.5 Schematic band structures on the K -H and K  -H  lines for a the ASOC α > 0 and b α < 0. The Bloch wave functions for all bands are shown. The red frame indicates the wave functions whose energy bands cross the Fermi level. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society

77

(a) K - H line

K’ - H ’ line

K - H line

K’ - H ’ line

(b)

Fig. 4.6 The phase factor ei k·r on sites for the K -point normal Bloch state. The sublattices a and b acquire different phase values by threefold rotation. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society

on the K -point, which results in λz = −1 (+1) for the a (b) sublattice. For a more general analysis of the effective orbital angular momentum, see Appendix of the Chapter.

78

4 Superconducting Gap Classification on High-Symmetry Lines

Table 4.5 The total angular momentum of the Bloch state. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society K -H line K  -H  line lz sz λz jz lz sz λz jz |a, ↑ |a, ↓ |b, ↑ |b, ↓

0 0 0 0

+1/2 −1/2 +1/2 −1/2

+1 +1 −1 −1

+3/2 +1/2 −1/2 −3/2

0 0 0 0

+1/2 −1/2 +1/2 −1/2

−1 −1 +1 +1

−1/2 −3/2 +3/2 +1/2

Using the above argument, the total angular momentum of the normal Bloch state can be calculated by jz = l z + sz + λz . In the |a, ↑ state on the K -H line (see Fig. 4.5a), for example, we obtain l z = 0, sz = +1/2, and λz = +1, namely, jz = +3/2. The total angular momenta of all normal Bloch states on the K -H and K  -H  lines are listed in Table 4.5. According to Fig. 4.5a and b, and Table 4.5, we identify the representations of the normal Bloch states crossing the Fermi level as follows. (a) α > 0: E 3/2 representation, because     |a, ↑ =  jz = + 23 , |b, ↓ =  jz = − 23 , on the K -H line, and     |b, ↑ =  jz = + 23 , |a, ↓ =  jz = − 23 , on the K  -H  line. (b) α < 0: E 1/2 representation, because     |a, ↓ =  jz = + 21 , |b, ↑ =  jz = − 21 , on the K -H line, and     |b, ↓ =  jz = + 21 , |a, ↑ =  jz = − 21 , on the K  -H  line. Assuming the E 2u order parameter, therefore, the gap classification in Eq. (4.10) indicates that the quasiparticle spectra on the K -H and K  -H  lines are point-nodal when α > 0, while they are gapped otherwise. Therefore, we have clarified that jz of the normal Bloch states and resulting gap structures depend on the sign of the ASOC α. In the next subsection, we demonstrate such unconventional gap structures by a numerical analysis of the model.

4.3 UPt3 (Space Group: P63 /mmc)

79

4.3.5 Gap Structures Depending on Bloch-State Angular Momentum Now we demonstrate unusual jz -dependent gap structures using a numerical calculation of our microscopic model. In order to investigate the superconducting gap structures, we take into account two-component order parameters in the E 2u IR of point group D6h : ˆ (4.18) (k) = η1 ˆ 1E2u + η2 ˆ 2E2u . The two-component order parameters are parameterized as  (η1 , η2 ) = (1, iη)/ 1 + η2 ,

(4.19)

with a real variable η. With respect to the basis functions ˆ 1E2u and ˆ 2E2u , we obtain the following equations by adopting the neighboring Cooper pairs in the crystal lattice of uranium ions: ˆ 1E2u = δ1 { px(intra) (k)sx − p (intra) (k)s y }σ0 y σ+ σ− + δ2 { px(inter) (k)∗ sx − p (inter) + δ2 { px(inter) (k)sx − p (inter) (k)s y } (k)∗ s y } y y 2 2

ˆ 2E2u

(4.20) + f (x 2 −y 2 )z (k)sz σx − d yz (k)sz σ y is y , = δ1 { p (intra) (k)sx + px(intra) (k)s y }σ0 y σ+ σ− + δ2 { p (inter) (k)sx + px(inter) (k)s y } (k)∗ sx + px(inter) (k)∗ s y } + δ2 { p (inter) y y 2 2

+ f x yz (k)sz σx − dx z (k)sz σ y is y ,

(4.21)

which are composed of the intra-sublattice p-wave, inter-sublattice p-wave, and inter-sublattice d + f -wave components given by px(intra) (k) =



eix sin k · ei ,

(4.22)

i

p (intra) (k) = y



y

ei sin k · ei ,

(4.23)

i

√ k z  x i k ·r i px(inter) (k) = −i 3 cos r e , 2 i i √ k z  y i k ·r i p (inter) (k) = −i 3 cos r e , y 2 i i √ k z  x i k ·r i dx z (k) = − 3 sin Im ri e , 2 i

(4.24) (4.25) (4.26)

80

4 Superconducting Gap Classification on High-Symmetry Lines

Table 4.6 Range of the parameter η in the A, B, and C phases of UPt3 [5, 6, 20, 21, 25, 26]. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society A phase B phase C phase

|η| = ∞ 0 < |η| < ∞ |η| = 0

√ k z  y i k ·r i d yz (k) = − 3 sin Im ri e , 2 i √ k z  x i k ·r i f x yz (k) = − 3 sin Re ri e , 2 i √ k z  y i k ·r i f (x 2 −y 2 )z (k) = − 3 sin Re ri e . 2 i

(4.27) (4.28) (4.29)

Pauli matrices in the spin and sublattice spaces are denoted by si and σi , respectively. σ+ and σ− are defined by σ± = σx ± iσ y . A similar model was introduced for an investigation of topological superconductivity in UPt3 [25, 26]. The model has also been used in the study of the polar Kerr effect [44] and the odd-frequency Cooper pairs [45]. In these previous studies, the inter-sublattice p-wave component in ˆ 1E2u and ˆ 2E2u was neglected. Here we take into account the component and show that it indeed plays an important role for the jz -dependent point node, because the other components vanish on the K -H line. We here assume that the d + f -wave component is dominant among all the order parameters since the purely f -wave state reproduces the multiple superconducting phase diagram illustrated in Fig. 4.2 [15–17, 20, 21]. On the other hand, an admixture of p-wave components allowed by symmetry changes the gap structure. Thus, we adopt small intra- and inter-sublattice p-wave components with 0 < |δ1 | 1 and 0 < |δ2 | 1, respectively. Now we revisit the multiple superconducting phases in Fig. 4.2 [15–17, 20, 21] from the viewpoint of the order parameter. The A, B, and C phases are characterized by the ratio of two-component order parameters η = η2 /iη1 , which is summarized in Table 4.6. A pure imaginary ratio of η1 and η2 in the B phase indicates a chiral superconducting state, which maximally gains the condensation energy. Indeed, a recent theoretical study based on our two-sublattice model [44] has elucidated a polar Kerr effect, which is in agreement with the experiment [24]. Owing to the p-wave components, the B phase is a nonunitary state. On the other hand, the TRS preserving A and C phases have been considered to be stabilized by weak symmetry breaking of the hexagonal structure, possibly induced by weak antiferromagnetic order [20, 21, 46, 47]. We here assume that the A phase is the ˆ 2E2u state (η = ∞), while the C phase is the ˆ 1E2u state (η = 0), and we take η non-negative without loss of generality. Superconducting gap structures on the K -H line can be obtained by diagonalizing the BdG Hamiltonian [Eq. (4.12)] including the two-component order parameters [Eqs. (4.18)–(4.21)]. Figure 4.7a and b represent the calculated quasiparticle energy

E/Δ

4.3 UPt3 (Space Group: P63 /mmc)

4 3 2 1 0 -1 -2 -3 -4

81

(b) α < 0

(a) α > 0

-3 -2 -1

0 kz

1

2

3

-3 -2 -1

0 kz

1

2

3

Fig. 4.7 Bogoliubov quasiparticle energy dispersion on the K -H line for a the ASOC α = 0.2 > 0 and b α = −0.2 < 0. A TRS preserving superconducting phase, namely the A phase (η = ∞) or C phase (η = 0), is assumed. The other parameters are set to be (t, tz , t  , μ, , δ1 , δ2 ) = (1, −1, 0.4, −5.2, 0.5, 0.04, 0.2) so that the K -FSs of UPt3 are reproduced. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society

dispersion in the A phase, which has a point-nodal (gapped) structure in the ASOC α is positive (negative). Qualitatively the same results are obtained in the C phase. All the results are consistent with the group-theoretical gap classification in Eq. (4.10). Thus, it is confirmed that the gap structures depending on the angular momentum jz of the normal Bloch state are realized on the K -FSs of UPt3 . Here we discuss the effects of SSB in the superconducting phase. As mentioned at the end of Sect. 4.3.2, group-theoretical gap classification does not take into account SSB in the ordered state. Therefore, the results of gap classification [Eq. (4.10)] cannot be applied to the SSB phase in a straightforward way. On the other hand, the spontaneous TRS breaking is taken into account in the topological gap classification [Eq. (4.11)], where C3 rotation symmetry remains. We also demonstrate such a TRS breaking case by using the effective model. The TRS breaking and C3 preserving order parameter (1 E u or 2 E u ) is realized for |η| = 1 (see the dashed line in Fig. 4.2), where a topological phase transition occurs [26]. Therefore we diagonalize the BdG Hamiltonian for η = 1, which results in the quasiparticle energy dispersion in Fig. 4.8. When the ASOC α is positive, both bands create nodes at the zero energy (Fig. 4.8a). For a negative α, on the other hand, one band holds nodes but the other one is fully gapped (Fig. 4.8b), which indicates a perfectly spin-polarized nonunitary state. For both IRs, the nodes on the K -H line are topologically characterized by the number of occupied states of the block-diagonalized Hamiltonian, which is obtained by using the threefold rotation matrix Uˆ BdG (C3 ). In this subsection, we have clarified the jz -dependent gap structures corresponding to the sign of the ASOC term in the effective model [Eq. (4.12)]. The remaining question is which representation is actually realized in UPt3 . Ab initio band-structure calculation shows that the normal Bloch state on the K -H line belongs to the E 1/2 representation [48]. Combining the first-principles calculation and the gap classification theory, we conclude that the superconducting gap opens on the K -H line in UPt3 except for the topologically-protected point nodes emerging in the B phase.

4 Superconducting Gap Classification on High-Symmetry Lines

E/Δ

82

4 3 2 1 0 -1 -2 -3 -4

(a) α > 0

-3 -2 -1

(b) α < 0

0 kz

1

2

-3 -2 -1

3

0 kz

1

2

3

Fig. 4.8 Bogoliubov quasiparticle energy dispersion on the K -H line for a the ASOC α = 0.2 > 0 and b α = −0.2 < 0. We assume the C3 preserving and TRS breaking B phase (η = 1). The other parameters are same as those in Fig. 4.7. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

4.3.5.1

Weyl Superconductivity in B Phase

We also calculate the nodal structure in the B phase for a general parameter 0 < η < ∞. In a TRS breaking state, there generally exists Weyl nodes characterized by a topological Weyl charge, which is defined by a monopole of Berry flux, qi =

1 2π

 d k · F(k).

(4.30)

∂k j u n (k)|∂kk u n (k),

(4.31)

S

Here, the Berry flux defined by Fi (k) = −i i jk

 E n (k) 5). With an influence from the gap zeros, Weyl nodes appear in the comparatively wide range of the B phase: 0.8  η  3. On the other hand, there is no node in the A and C phases for jz = ±1/2 bands (Fig. 4.9b, η < 0.9 and η > 1.5). Therefore the B phase hosts Weyl nodes only in the narrow region around the η = 1 line in Fig. 4.2: 0.95  η  1.1. As a result, reflecting the angular momentum dependence of the gap classification on the C3v -symmetric K -H line, the structures of Weyl nodes are obviously different between jz = ±3/2 (Fig. 4.9a) and jz = ±1/2 (Fig. 4.9b). The nodal structure of Weyl superconductors may be clarified by the thermal Hall conductivity [52].

4.4 SrPtAs (Space Group: P63 /mmc) 4.4.1 Background SrPtAs is a pnictide superconductor with a hexagonal crystal structure characterized 4 ). First-principles calculations by a nonsymmorphic space group P63 /mmc (D6h based on local density approximation indicate the presence of 2D FSs enclosing the -A line, a 2D FS enclosing the K -H line, and a 3D FS crossing the K -H line [53– 55]. Pairing symmetry in SrPtAs is still under debate owing to the inconsistency of some experiments: for example, a TRS breaking and nodeless superconducting state suggested in a muon spin-rotation/relaxation measurement [56] is incompatible with a spin-singlet s-wave superconductivity with an isotropic gap indicated by recent 195 Pt-NMR and 75 As-NQR measurements [57]. In theoretical studies, on the other hand, an E 2g state with a chiral d-wave order parameter [58–60] and a B1u state with an f -wave pairing [61, 62] have been pro-

84

4 Superconducting Gap Classification on High-Symmetry Lines

(a)

η < 0.2, η > 5

η = 0.8

q = −2

η =1

q = +1

K q = −1 q = +2

η = 1.5

η=3

kz ky

(b)

η < 0.9, η > 1.5

η = 0.95

η = 1.05

η = 1.1

η =1

kz ky Fig. 4.9 Illustration of pair creation and annihilation of Weyl nodes on the K -FS for a jz = ±3/2 (α > 0) and b jz = ±1/2 (α < 0). Small blue and red circles show single Weyl nodes with qi = 1 and −1, respectively. Large circles are double Weyl nodes with qi = ±2. Orange circles are topologically trivial point nodes protected by C3 symmetry. Closed (open) circles represent nodes on the front (back) side of the FS. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

4.4 SrPtAs (Space Group: P63 /mmc)

85

(a1) jz = ± 3 / 2, η < 1

(a2) jz = ± 3 / 2, η > 1

5

5

4

4

ν(kz)

3

3

η = 0.2 η = 0.6 η = 0.8 η=1

2 1

2 1

0

0

-1

-1

-2

-2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

5

5

4

4

η = 0.4 η = 0.95 η = 0.98 η=1

1

0.2

0.3

0.4

0.5

0.6

0.7

0.5

0.6

0.7

η = 1.5 η = 1.1 η = 1.05 η = 1.02 η=1

3

3 2

0.1

(b2) jz = ± 1 / 2, η > 1

(b1) jz = ± 1 / 2, η < 1

ν(kz)

η=6 η=3 η = 1.5 η=1

2 1

0

0

-1

-1 -2

-2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kz

0

0.1

0.2

0.3

0.4 kz

Fig. 4.10 A k z -dependent Chern number defined by Eq. (4.32) on each 2D slice for a jz = ±3/2 (α > 0) and b jz = ±1/2 (α < 0). The parameters are set to be (t, tz , t  , α, μ, , δ1 , δ2 ) = (1, −1, 0.4, ±0.2, −5.2, 0.1, 0.04, 0.2) so that the K -FSs are reproduced. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

posed besides the fully-gapped A1g order parameter suggested by the NMR measurement [57]. The following discussion is mainly based on the chiral E 2g superconductivity [58–60], which is compatible with broken TRS [56] and the decrease of spin susceptibility below Tc [57], although it is inconsistent with the nodeless gap structure [56, 57]. We particularly focus on the gap structure on the C3v -symmetric K -H line (see Fig. 4.4), since recent theoretical studies have suggested the presence of Weyl nodes [59] or Bogoliubov FSs [8, 9, 63] on the axis.

4.4.2 Classification on K -H Line Now we consider the gap structure on the K -H line with the C3v symmetry, which has been already classified in Eq. (4.10). Since the E 2g IR is prohibited for both λ¯ αk = E 1/2 and E 3/2 in Eq. (4.10), the chiral d-wave superconductivity hosts nodes on the K -H line. As described above, this contradicts the nodeless superconductivity suggested in Refs. [56, 57]. The B1u state is consistent with the nodeless gap structure, if the normal Bloch state on the K -H line is categorized as E 3/2 , although the 1D IR is inconsistent with broken TRS. The E 1u and E 2u superconductivity for the E 1/2 normal Bloch state

86

4 Superconducting Gap Classification on High-Symmetry Lines

are consistent with both the nodeless gap structure and broken TRS. However, these odd-parity superconducting states have an inconsistency with the NMR Knight shift measurement, which indicates the decrease of spin susceptibility below Tc [57]. Here, we discuss the topological classification of the E 2g pairing state. The compatibility relation reveals that E 2g of D6h corresponds to E g of D3d . According to Table 4.2, E g gap closes on the line irrespective of the angular momentum of normal Bloch states, as mentioned in the previous paragraph. However, Table 4.3b1 and b2 shows the distinct topological classifications of such nodes depending on the angular momentum: Z2 ⊕ Z for α = ±1/2, Z ⊕ Z for α = ±3/2.

(4.34a) (4.34b)

Next, we identify the concrete formulations of the topological numbers. Without loss of generality, we fix symmetry of the superconducting order parameter to the 2 E g IR, which acquires a U(1) phase with +2π/3 under a C3 rotation: ˆ + (k)C3T = e+i2π/3  ˆ + (k), C3 

(4.35)

for a k point on the K -H line (see also Table 4.4a). The choice of the TRS breaking (chiral) order parameter is same as that in previous theoretical studies [58–60]. Although the only intrinsic symmetry preserving any k point is the pseudo-PHS C in the chiral superconducting state, the pseudo-TRS T and the CS  recover by considering a hypothetical 0 → 0 limit (0 : the magnitude of the order parameter). Thus, in this limit, the minimal BdG Hamiltonian on the C3 -symmetric line is defined by Eq. (2.45) and C † (k) = (cα† (k), Tcα† (k)T−1 , Ccα† (k)C−1 , cα† (k) −1 ),

(4.36)

for α = +1/2 or +3/2 reflecting the property of the normal Bloch state. The BdG Hamiltonian matrix is described by   0 ψ(k)iσ y ξ(k)σ0 − h(k)20 σz , Hˆ BdG (k) = (0 ψ(k)iσ y )† −(ξ(k)σ0 − h(k)20 σz )

(4.37)

where h(k)20 is a “pseudomagnetic” field representing the spontaneous TRS breaking in the superconducting ordered state. In real multi-band systems, such a field arises from a second-order perturbation of the inter-band pairing [9, 64], and cannot be neglected in general. Note that the pseudomagnetic field does not break C3 symmetry since its direction is parallel to the C3 -symmetric axis. We discuss the topological numbers (Z2 ⊕ Z) for α = ±1/2 bands. Reflecting the C3 symmetry, the BdG Hamiltonian matrix commutes with the threefold rotation matrix Uˆ BdG (C3 ): (4.38) [ Hˆ BdG (k), Uˆ BdG (C3 )] = 0,

4.4 SrPtAs (Space Group: P63 /mmc)

87

which shows that Hˆ BdG (k) and Uˆ BdG (C3 ) can be simultaneously block-diagonalized, that is, there exists a unitary matrix Vˆ such that ⎛ Hˆ BdG (k) = Vˆ

Uˆ BdG (C3 ) = Vˆ

⎞ Hˆ e+iπ/3 (k) 0 0 ⎝ 0 Hˆ e−iπ/3 (k) 0 ⎠ Vˆ † , 0 0 Hˆ −1 (k) ⎛ +iπ/3 ⎞ e 12 0 0 ⎝ 0 e−iπ/3 0 ⎠ Vˆ † . 0 0 −1

(4.39)

(4.40)

The Hamiltonian blocks He+iπ/3 (k), He−iπ/3 (k), and H−1 (k) are described by 1 † † (k), Cc+1/2 (k)C−1 ) He+iπ/3 (k) = (c+1/2 2    c+1/2 (k) 0 ξ(k) − h(k)20 , × 0 −(ξ(k) − h(k)20 ) Cc+1/2 (k)C−1 1 † He−iπ/3 (k) = Tc+1/2 (k)T−1 (ξ(k) + h(k)20 )Tc+1/2 (k)T−1 , 2 1 † H−1 (k) = c+1/2 (k) −1 (−ξ(k) − h(k)20 )c+1/2 (k) −1 . 2

(4.41) (4.42) (4.43)

Band structures derived from the Hamiltonian (4.39) are schematically shown in Fig. 4.11a. The C3 eigenvalues of the four bands can be identified by considering the 0 → 0 limit and the property of T, C, and . Even when 0 is finite, i.e., the TRS T and the CS  are broken, the eigenvalues are not changed since the commutation relation [Eq. (4.38)] remains preserved. Then, the band degeneracy splits because of the pseudomagnetic field h(k)20 , and nodes emerge at the zero energy because the spin-singlet 2 E g gap function cannot have offdiagonal components in the same eigenspace. The nodes are categorized into two types, one of which is on the α = −1/2 particle band (the pink one in Fig. 4.11a), and the other is on the α = +1/2 particle band (the blue one in Fig. 4.11a). The former nodes are obviously protected by a Z number, (4.44) ν− ≡ #(occupied states of Hˆ e−iπ/3 (k)) ∈ Z. On the other hand, the latter ones cannot be characterized by a similar filling formula, since a hole band belonging to the same eigenspace of C3 simultaneously exists (see the e+iπ/3 space in Fig. 4.11a). Since the eigenspace is closed under the (pseudo-)PHS C with C2 = +E, the nodes are instead protected by the Z2 index [8, 65], (−1)l+ ≡ sgn[i n Pf{Uˆ C,e+iπ/3 Hˆ e+iπ/3 (k)}] ∈ Z2 , with n = dim( Hˆ e+iπ/3 )/2.

(4.45)

88

4 Superconducting Gap Classification on High-Symmetry Lines

(a)

(b)

Fig. 4.11 Band-theoretical picture of BdG Hamiltonian with an 2 E g order parameter on a C3 † in a and b is an abbresymmetric line for a α = ±1/2 and b α = ±3/2 normal Bloch states. c+ † † viated notation of c+1/2 (k) and c+3/2 (k), respectively. In the right panels, nodes on the line are depicted by the red points. Although the particle bands and the hole bands in the limit 0 → 0 are doubly degenerated, both of them split for a finite 0 due to a spontaneous TRS breaking in the superconducting state. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

The topological numbers (Z ⊕ Z) for α = ±3/2 bands are obtained in the following. In this case, the simultaneous block-diagonalization of the BdG Hamiltonian matrix and the threefold rotation matrix is written by  0 Hˆ −1 (k) Vˆ † , 0 Hˆ e−iπ/3 (k)   0 −12 ˆ ˆ UBdG (C3 ) = V Vˆ † , 0 e−iπ/3 12 Hˆ BdG (k) = Vˆ



(4.46) (4.47)

where 1 † † (k), Tc+3/2 (k)T−1 ) H−1 (k) = (c+3/2 2    0 c+3/2 (k) ξ(k) − h(k)20 , × 0 ξ(k) + h(k)20 Tc+3/2 (k)T−1

(4.48)

4.4 SrPtAs (Space Group: P63 /mmc)

1 † † He−iπ/3 (k) = (Cc+3/2 (k)C−1 , c+3/2 (k) −1 ) 2    0 Cc+3/2 (k)C−1 −(ξ(k) − h(k)20 ) . × c+3/2 (k) −1 0 −(ξ(k) + h(k)20 )

89

(4.49)

Figure 4.11b schematically illustrates band structures obtained by the BdG Hamiltonian matrix (4.46). Contrary to the α = ±1/2 case, both of the α = ±3/2 particle bands (the green ones in Fig. 4.11b) belong to the same C3 eigenspace with an eigenvalue −1. Therefore, all nodes on the C3 -symmetric line are characterized by a filling number, (4.50) ν± ≡ #(occupied states of Hˆ −1 (k)) ∈ Z. Regardless of the IR α of the normal Bloch state, the chiral even-parity 2 E g order parameter induces the splitting of band degeneracy and the emergence of nodes on the C3 -symmetric K -H line. These facts show that the nodes are parts of inflated Bogoliubov FSs, which are protected by a 0D topological number defined on any k point, namely the Pfaffian of the antisymmetrized BdG Hamiltonian, P(k) ∈ Z2 [8]. Indeed, the existence of Bogoliubov FSs in SrPtAs has been suggested in previous studies [8, 9, 63]. In addition, our result in Eq. (4.34) finds that the topological protection of the nodes on the K -H line is differently characterized reflecting the property of the IR α. Although the above intuitive discussions are based on the single-band model in the weak coupling limit, the topological protection ensures the stability of nodes against multi-band effects.

4.5 CeCoIn5 (Space Group: P4/mmm) 4.5.1 Background A Ce-based heavy-fermion compound CeCoIn5 has a tetragonal crystal structure 1 ). Angle-resolved photoemiswhich is characterized by a space group P4/mmm (D4h sion spectroscopy studies [66, 67], dHvA measurements [68–71], and first-principles calculations [72, 73] have suggested the presence of 3D FSs crossing the C4v symmetric -Z line in the compound. CeCoIn5 shows superconductivity at ambient pressure below 2.3 K [74]. In terms of the pairing symmetry, the presence of line nodes is indicated by specific heat and thermal conductivity measurements [75], and NQR relaxation rate 1/T1 [76]. Furthermore, scanning tunneling spectroscopy [77– 79], field-angle-resolved measurements of thermal conductivity [80] and heat capacity [81], and torque magnetometry [82] strongly support the dx 2 −y 2 -wave (B1g ) symmetry of the superconducting order parameter.

90

4 Superconducting Gap Classification on High-Symmetry Lines

4.5.2 Classification on -Z Line We classify superconducting gap structure on the -Z line with the C4v symmetry. The group-theoretical classification (Table 4.2) elucidates that the gap structure on the axis is nodal for the dx 2 −y 2 -wave order parameter belonging to the B1g IR of D4h , irrespective of the total angular momentum jz of the normal Bloch state. The nodes on the axis are parts of a line node because of the following reason. In the tetragonal BZ, the -Z -A-M plane, which of course includes the -Z line, is invariant under the diagonal mirror M[110] . A gap structure on the plane is classified in Table 3.1, P¯ k = A g + 2 Au + Bu ,

(4.51)

[110] . The superscript [110] means which is decomposed by IRs of the point group C2h that the principal axis is parallel not to the z direction but to the [110] direction. [110] , Therefore the compatibility relation between the original point group and C2h [110] = Bg , (B1g of D4h ) ↓ C2h

(4.52)

shows that a line node appears on the [110] plane.5 A 1D winding number Wl ∈ Z [Eq. (3.12)], which is defined on a loop l encircling the nodal line using the CS , characterizes the line node [11, 12]. Thus the node is stable even when the mirror symmetry M[110] is broken as long as the CS is preserved. In general, nodes on a high-symmetry axis may be parts of line nodes when a superconducting order parameter is even-parity and preserves TRS [see (L) in Table 4.3]. According to Table 4.3g, the topological classification on the fourfold -Z line for the B1g order parameter is the class AI (Z). The orthogonality test for the CS  is Wα = 0, which indicates that  maps bases of the IR α = 1/2 (3/2) to those of α  = 3/2 (1/2). Therefore the Z index is defined by the number of occupied states of the Hamiltonian block in the eigenspace of the IR α.

4.6 UCoGe (Space Group: P nma) 4.6.1 Background UCoGe is an orthorhombic superconductor whose crystal structure is categorized to 16 a nonsymmorphic space group Pnma (D2h ) [83]. In this material, the coexistence of superconductivity and ferromagnetism has been observed at ambient pressure [84, 85], which strongly suggests an intrinsic odd-parity superconductivity. The highpressure superconducting phase [84–90] (S2 phase in Ref. [87]), where TRS recovers by the vanishing FM moments [86–88, 91], is also expected to be odd-parity 5 The

same result is obtained for the [1−10] plane.

4.6 UCoGe (Space Group: Pnma)

91

superconductivity, since it is continuously connected to the FM superconducting phase [86–90]. Although the pairing symmetry of UCoGe is still under debate, recent studies have suggested that UCoGe is a plotform for topological crystalline superconductivity, which is characterized by Z4 and/or Z2 numbers, for all possible odd-parity superconducting order parameters (Au , B1u , B2u , and B3u IRs) [1, 92]. Especially, point nodes appear on the -Z line when the order parameter belongs to the B1u IR [1]. In the discussion below, therefore, we focus on the B1u order parameter as a candidate of the odd-parity superconducting order.

4.6.2 Classification on -Z Line We classify gap structures, in the high-pressure PM superconducting phase of UCoGe, on the -Z line with C2v symmetry. According to Table 4.2, the grouptheoretical gap classification is P¯ k = A g + Au + B2u + B3u , which indicates the presence of point nodes on the axis in the B1u superconducting state. On the other hand, the topological classification shows that the point nodes are protected by a Z2 number (see Table 4.3e). The Z2 topological index is identified as follows. The IR of the little cogroup G¯ k ∼ = C2v is written by k (E) = σ0 , γ¯1/2 k γ¯1/2 (C2z ) k γ¯1/2 (M y )

(4.53a)

= −iσz ,

(4.53b)

= −iσ y ,

(4.53c)

k γ¯1/2 (Mx ) = −iσx ,

(4.53d)

† where the bases [c±i (k)] are the eigenstates of C2z with the eigenvalues ±i. Thus the † † minimal BdG Hamiltonian is expressed by the four bases, c±i (k) and Cc±i (k)C−1 . Here, we remark that the PHS C changes the eigenvalue of C2z since C is an antiunitary operator with [C, C2z ] = 0 in the B1u superconducting state. Furthermore, Eq. (4.53) shows that the mirror operator M y also changes the eigenvalue of C2z . Therefore a new PHS operator  C ≡ CM y preserves the eigenvalue of C2z , and it has the following relation,  (4.54) C2 = C2 M2y = +E,

where we use [C, M y ] = 0 for the B1u order parameter. As a result the BdG Hamiltonian matrix Hˆ BdG (k) are decomposed by C2z eigen-sectors Hˆ ± (k), each of which has the pseudo-PHS operator  C. Due to the symmetry, the Z2 number can be defined in each sector [8, 65]: ˆ (−1)l± ≡ sgn[i n Pf{Uˆ  C,± H± (k)}] ∈ Z2 ,

(4.55)

4 Superconducting Gap Classification on High-Symmetry Lines

E/Δ

4

+i sector

2

1

0

0

-2

-1

(-1)l+

92

-4 -3 -2 -1

0 kz

1

2

3

Fig. 4.12 The quasiparticle energy dispersion (blue lines) and the Z2 topological number (red line) on the -Z line for the +i sector of the block-diagonalized BdG Hamiltonian. The result for the −i sector is the same as the +i case. We adopt the effective model in Refs. [1, 92]. The parameters  , μ, t  , α) = (1, 0.2, 0.1, 0.5, 0.1, −0.5, 0.1, 0.3) are assumed so that the -FSs (t1 , t2 , t3 , tab , tab 1 of UCoGe are reproduced. Reprinted with permission from Ref. [6]. Copyright © 2019 by the American Physical Society

with n = dim( Hˆ ± )/2. We demonstrate the topological crystalline point nodes by using the effective four-sublattice single-orbital model introduced in Refs. [1, 92]. Figure 4.12 shows the eigenvalues of the BdG Hamiltonian and the Z2 index for the +i eigen-sector on the -Z line. Obviously, we find that the Z2 topological number (−1)l+ changes at the point nodes.

4.7 MoS2 (Space Group: P63 /mmc) 4.7.1 Background MoS2 is a member of the group-VI transition-metal dichalcogenides MX 2 (M = Mo, W; X = S, Se, Te). In the compound, superconductivity has been observed not only in a ion-gated atomically thin 2D system [39, 93–96], but also in a bulk intercalated system [97, 98]. In these electron-doped systems, the major contribution in the spin-split lowest conduction bands, which form FSs around the K point, is Mo-dz 2 orbitals [99]. However, the FSs do not intersect with the K -H axis since the lowest bands are almost dispersionless along the axis. That is attributed to the absence of the nearest inter-layer hopping [42]. On the other hand, Mo-dx 2 −y 2 ± idx y orbitals contribute to the spin-split top valence bands [99], which have sizable dispersion on the K -H line in the 2H stacking structure [42]. Thus, it may be possible that these top valence bands form FSs crossing the K -H axis in a hole-doped MoS2 .

4.7 MoS2 (Space Group: P63 /mmc)

93

4.7.2 Classification on K -H Line Since the crystal symmetry of bulk MoS2 is a hexagonal space group P63 /mmc, superconducting gap structures on the K -H line is already classified by Eq. (4.10). The detailed result of the gap classification is discussed below. Although the symmetry of a superconducting order parameter in MoS2 has not been determined, a recent theoretical study [100] has suggested a conventional BCS superconductivity (A1g ) in the PM regime, and a pair-density-wave state (B2u ) under the external magnetic field. The former A1g symmetry is supported by recent first-principles calculations, where electron-phonon interactions are taken into account [101–103]. A gapped structure appears on the K -H line for both A1g and B2u superconductivity, since Eq. (4.10) contains both IRs irrespective of the angular momentum of the normal Bloch state. On the other hand, other theoretical studies have proposed topological superconductivity in electron-doped [104] and hole-doped [105] monolayer MoS2 , where the pairing symmetry is classified into A1 or E IR of a point group C3v . Note that the point group symmetry of monolayer MoS2 is C3v , while that of bulk is D6h . Thus for the bulk material, these IRs are induced to D6h as A1 ↑ D6h = A1g + B2g + A2u + B1u , E ↑ D6h = E 1g + E 2g + E 1u + E 2u ,

(4.56) (4.57)

where we assume that the pairing symmetry in the bulk is the same as that in the monolayer. Therefore, the choice of the basis function of the A1 or E IR determines the presence or absence of point nodes on the K -H line. The detailed result is summarized in Table 4.7, which indicates that the superconducting gap structure

Table 4.7 Gap structure on the K -H axis in bulk MoS2 where the pairing symmetry belongs to A1 or E IR of C3v . “PS” in the first and second columns represents pairing symmetry. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society PS in C3v PS in D6h Normal Bloch state Gap structure A1

A1g B2g A2u B1u

E

E 1g E 2g E 1u E 2u

E 1/2 , E 3/2 E 1/2 , E 3/2 E 1/2 E 3/2 E 1/2 E 3/2 E 1/2 , E 3/2 E 1/2 , E 3/2 E 1/2 E 3/2 E 1/2 E 3/2

Gap Gap Point node Gap Point node Gap Point node Point node Gap Point node Gap Point node

94

4 Superconducting Gap Classification on High-Symmetry Lines

depends on the effective angular momentum jz when the pairing symmetry is A2u , B1u , E 1u , or E 2u .

¯ 4.8 UBe13 (Space Group: Fm3c) 4.8.1 Background Superconductivity in UBe13 was discovered in 1983 [106]. However the nature and the symmetry of superconductivity are still under debate. For example, previous experiments have proposed a point-nodal p-wave [107] and line-nodal [108] superconductivity, while a fully opened gap structure has suggested by more recent angleresolved heat-capacity measurements [109]. Furthermore, another mystery about UBe13 is the emergence of a second phase transition inside the superconducting state when a small amount of U atoms are replaced by Th [110, 111]; μSR [112] and thermal-expansion [113] experiments have reported the existence of four superconducting phases.

4.8.2 Classification on -L Line The crystal structure of UBe13 is characterized by a face-centered cubic space group ¯ where the BZ has a threefold rotation axis -L (see Fig. 4.13a). Although ab Fm 3c, initio calculations indicate only a tiny FS intersecting with the -L line [114, 115], the FS structure has not been confirmed by experiments. In this subsection, therefore, a gap classification on the -L line is carried out assuming the existence of FSs in the [111] direction. The little cogroup on the -L line is G¯ k ∼ = C3v , which results in two

(a)

(b) P

L Γ

H Γ

Fig. 4.13 The first BZ of a face-centered cubic lattice and b body-centered cubic lattice. The red lines show threefold rotation symmetric axes. Reprinted with permission from Ref. [5]. Copyright © 2018 by the American Physical Society

¯ 4.8 UBe13 (Space Group: Fm 3c)

95

different representations of the Cooper pair wave function [Eq. (4.9)], corresponding to two IRs λ¯ k = E 1/2 and E 3/2 of the normal Bloch state. By considering an induction of P¯ k in Eq. (4.9) to the original crystal point group Oh , the group-theoretical gap classification in UBe13 is obtained:  A1g + T2g + A1u + E u + T1u + 2T2u (λ¯ k = E 1/2 ), k P¯ ↑ Oh = (4.58) A1g + T2g + 2 A1u + A2u + T1u + 2T2u (λ¯ k = E 3/2 ). In this superconductor, the E u state [116, 117] and the accidental A1u + A2u mixed state [116], which are consistent with the double transition in U1−x Thx Be13 , have been proposed. A jz -dependent gap structure appears on the -L line in the former E u state: the superconducting gap opens for λ¯ k = E 1/2 , while point nodes emerge for λ¯ k = E 3/2 [see Eq. (4.58)]. Previous studies based on the E u scenario simply have suggested the fully gapped state in the TRS preserving A and C phases, since they assumed the E 1/2 normal Bloch state [116]. However, their results are not valid when the E 3/2 normal Bloch state contributes to the FS crossing the -L axis. In the latter accidentally mixed state, on the other hand, the A1u component opens the gap on the line irrespective of the angular momentum of the normal Bloch state. The other A2u component contributes the gap opening only for the E 3/2 normal Bloch state. For both E u and A1u + A2u scenarios, identifying the angular momentum of normal Bloch states is essential for elucidating the relationship between symmetry and superconducting ap structures. The experimental data should be carefully interpreted by taking into account this fact.

¯ 4.9 PrOs4 Sb12 (Space Group: I m3) 4.9.1 Background PrOs4 Sb12 is a heavy-fermion superconductor with the filled skutterudite structure RT 4 X 12 (R = rare earth or U; T = Fe, Ru, Os; X = P, As, Sb). The manifestation of unconventional superconductivity in PrOs4 Sb12 has been repoted in a lot of studies [118, 119]. For example, specific-heat [120, 121] and thermal transport [122] measurements have suggested multiple superconducting phases. However, the pairing symmetry in PrOs4 Sb12 remains controversial even now: some NQR [123], penetration depth [124], and specific heat [120, 121] studies have suggeted a point-nodal superconductivity, while a fully gapped superconductivity has been proposed in other thermal conductivity [125], μSR [126], and NQR [127] measurements. Furthermore, several experiments have observed TRS breaking in the low-temperature and lowmagnetic-field superconducting phase (B phase) [128, 129].

96

4 Superconducting Gap Classification on High-Symmetry Lines

4.9.2 Classification on - P and P-H Lines Here we perform a group-theoretical gap classification in PrOs4 Sb12 . The crystal lat¯ where threefold tice of PrOs4 Sb12 belongs to a body-centered cubic space group I m 3, rotation axes -P and P-H exist in the BZ (see Fig. 4.13b). The topology of FSs in PrOs4 Sb12 has been confirmed by the combination of dHvA experiment and firstprinciples calculation [130]. The determined FSs are composed of three parts, two of which intersect with the -P line and the other crosses the P-H line. Therefore, the gap structure on these lines is worth considering. The little cogroup on the -P and P-H lines is G¯ k ∼ = C3 , which has in two IRs λ¯ k = E 1/2 and 2B3/2 (Table 4.1b). Corresponding to these two normal Bloch states, two nonequivalent representations of the Cooper pair wave function are derived as shown in Table 4.2. Thus the induced representation P¯ k ↑ Th of the original point group Th is given by  ¯k

P ↑ Th =

A g + Tg + Au + E u + 3Tu (λ¯ k = E 1/2 ), (λ¯ k = 2B3/2 ). A g + Tg + 3Au + 3Tu

(4.59)

Theoretical studies have suggested various possibilities of pairing symmetry in PrOs4 Sb12 [131–136]. For example, the 3D Tg and Tu states [131], the mixed A g + E g state with a s + g-wave pairing [132], and that with a s + id-wave pairing [134] have been proposed. In these cases, a fully gapped structure appears on the -P and P-H lines irrespective of the angular momentum of the normal Bloch state. However, the presence or absence of point nodes on the axes may be jz -dependent, if the order parameter belongs to the E u IR [see Eq. (4.59)].

Appendix: Effective Orbital Angular Momentum Due to Sublattice Degree of Freedom In Sect. 4.3.4, the effective orbital angular momentum λz , which arises from the uranium sublattice degree of freedom, is introduced. This Appendix provides a general formulation for the angular momentum by considering transformation of a normal Bloch wave function. We introduce a sublattice-based creation operator of the normal Bloch state † (k), where m and ζ = l z + sz are indices of the sublattice and ordinary denoted by cmζ total angular momentum, respectively. Fourier transformation of the normal Bloch state is explicitly defined by † (k) = cmζ



e−i k·R cζ† (R + r m ),

(4.60)

R

where R shows a position of the unit cell origin, while r m is a relative position of the sublattice m in the unit cell. Using the equation, the creation operator is transformed

¯ 4.9 PrOs4 Sb12 (Space Group: I m 3)

97

by a space group operation g = { p|a} as † gcmζ (k)g −1 =



e−i k·R gcζ† (R + r m )g −1

R

=



e−i k·R

 ζ

R

cζ† ( p(R + r m ) + a)Dζ(Z ζ) ( p),

where D (Z ) ( p) is a representation matrix of p with a total angular momentum Z = l + s. Defining R + r pm ≡ p(R + r m ) + a, we obtain † (k)g −1 = gcmζ

 R

=e

e−i k·[ p



i pk·a

−1

(R +r pm −a− pr m )]

   ζ

R

=e

i pk·a



=e

i pk·a



ζ

m



 ζ

 e−i pk·R cζ† (R

cζ† (R + r pm )Dζ(Z ζ) ( p) 

+ r pm ) e−i pk·(r pm − pr m ) Dζ(Z ζ) ( p)

c†pm,ζ  ( pk)e−i pk·(r pm − pr m ) Dζ(Z ζ) ( p)

ζ

cm†  ζ  ( pk)[D(perm) ( p, k)]m  m Dζ(Z ζ) ( p).

(4.61)

According to Eq. (4.61), a representation matrix indicating the permutation of sublattices is defined by [D(perm) ( p, k)]m  m = e−i pk·(r pm − pr m ) δm  , pm .

(4.62)

The phase factor in the representation corresponds to an effective orbital angular momentum λz . For example, a threefold rotation in UPt3 is represented by

a D(perm) (C3 , k) = b



a b  e+i2π/3 0 , 0 e−i2π/3

(4.63)

on the K point [k = (4π/3, 0, 0)]. The phase factors in Eq. (4.63) give the effective angular momentum λz = ±1 as we demonstrated in Sect. 4.3.4.

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Chapter 5

Conclusion

In this thesis, we investigate gap structures in various unconventional superconductors with spin-orbit coupling and multi-degrees of freedom, by using the modern classification theory of superconducting gap nodes. In Chap. 2, we present the recently developed classification methods of superconducting gap, from both aspects of group theory and topology [1–4]. The methods take into account space group symmetry and higher-spin normal Bloch states, which enables us to reveal novel types of nodal structures beyond the order parameter analysis by the Sigrist-Ueda method. Moreover, the combination of group-theoretical and topological classifications offers the complementary understanding of nodes appearing in unconventional superconductors: the group-theoretical analysis helps us to easily search symmetry-protected superconducting nodes, while the property of topological number tells us the stability of such nodes. Our classification theory motivates the research community to reacknowledge the importance of such complementary studies, and to expect that all crystal symmetry-protected nodes are protected by topology. In Chap. 3, we completely classify gap structures on high-symmetry planes (mirror- or glide-invariant planes) in the Brillouin zone [3]. When the system has symmetry including non-primitive translation parallel to a twofold axis, the Cooper pair wave functions on the basal plane and the zone face, which are perpendicular to the twofold axis, have different representations as a consequence of the nonsymmorphic symmetry. In this case, therefore, line nodes (or a gap opening) protected by nonsymmorphic symmetry may emerge on the Brillouin zone boundary. We classified the gap structure of all the centrosymmetric space groups. From the list of space groups, we may understand the symmetry-protected line node for each crystal, magnetic, and superconducting symmetries. Furthermore, we have established the relationship between such symmetry-protected line nodes and Majorana flat bands using a topological argument [2]. The zero-dimensional topological number not only reflects the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Sumita, Modern Classification Theory of Superconducting Gap Nodes, Springer Theses, https://doi.org/10.1007/978-981-33-4264-4_5

103

104

5 Conclusion

group-theoretical results, but also relates to the one-dimensional topological number that ensures the existence of Majorana flat bands. By analyzing the relationship between the two topological numbers, we can categorize the symmetry-protected line nodes into three distinct classes. Each class is distinguished by the Majorana flat bands and the symmetry-protected line nodes may be distinguished through surface sensitive experiments such as tunneling conductance measurements. Finally, as an example, we suggest the nonsymmorphic symmetry-protected gap structures in the − + +− (magnetic octupole) state of Sr2 IrO4 [1]. We confirmed the symmetry-based classification results by the realistic model for Sr2 IrO4 . In Chap. 4, on the basis of the symmetry and topology we classified superconducting gap structure on high-symmetry n-fold lines in the Brillouin zone [3, 4]. First, we execute the comprehensive group-theoretical classification; on threefold and sixfold axes, the gap classification depends on the total angular momentum of the normal Bloch state jz , while that is unique on twofold and fourfold axes. Next, we find that the topological analysis completely corresponds with the above grouptheoretical classification; all nodes shown by group theory are characterized by a zero-dimensional Z or Z2 index. Thus the symmetry-protected nodes are topologically protected. Such topological crystalline superconducting nodes on the highsymmetry axes are classified into three types: point nodes, a part of line nodes, and a part of surface nodes (Bogoliubov Fermi surfaces). Furthermore, we applied such classification to various candidate superconductors: UPt3 , SrPtAs, CeCoIn5 , UCoGe, MoS2 , UBe13 , and PrOs4 Sb12 . Especially, we showed that the structure of Weyl nodes also depends on the angular momentum of the normal Bloch state in the time-reversal symmetry breaking B phase of UPt3 , reflecting the jz -dependent gap structure on the C3v -symmetric K -H line. Let us introduce recent developments and further outlooks around the gap classification theory. As a main development, we point out an uranium superconductor UTe2 discovered in the end of 2018 [5]. Since a lot of experiments have supported the presence of spin-triplet superconductivity, this compound has explosively attracted much attention. Stimulated by the background, we recently performed a detailed superconducting gap classification and discussed the possibility of topological superconductivity in UTe2 , based on band structures obtained by a first-principles calculation [6]. The gap classification method introduced in this thesis is useful for such a new material. Furthermore, we present outlooks of the study. This thesis offers almost complete classification of superconducting gap nodes on high-symmetry regions in the Brillouin zone. However, there are several remaining issues to be uncovered. First, gap classification on high-symmetry lines may be changed by taking nonsymmorphic symmetry into account, although only symmorphic symmetry is considered in Chap. 4. Indeed, Ref. [7] shows gap classification for the nonsymmorphic magnetic space group Pnma1 , which gives nontrivial gap structures on C2v -symmetric hinges of the Brillouin zone with glide symmetry. Secondly, it may be interesting to investigate the relationship among more than one topological numbers characterizing nodes on high-symmetry axes. In time-reversal symmetry breaking superconductors, for example, it may be possible to reveal such relationship between a two-dimensional

5 Conclusion

105

Chern number and a zero-dimensional topological number, which enables us to understand the stability of the Weyl nodes and the corresponding surface Majorana arc states. Finally, our classification theory can be extended to lower-dimensional superconductors and/or spin-orbit decoupled superconductors; such classification may be useful in determining superconducting symmetry of a twisted bilayer graphene, which has recently attracted much attention. The above ideas are interesting problems and have the potential to find new physical phenomena in future.

References 1. 2. 3. 4. 5.

Sumita S, Nomoto T, Yanase Y (2017) Phys Rev Lett 119:027001 Kobayashi S, Sumita S, Yanase Y, Sato M (2018) Phys Rev B 97:180504(R) Sumita S, Yanase Y (2018) Phys Rev B 97:134512 Sumita S, Nomoto T, Shiozaki K, Yanase Y (2019) Phys Rev B 99:134513 Ran S, Eckberg C, Ding QP, Furukawa Y, Metz T, Saha SR, Liu IL, Zic M, Kim H, Paglione J, Butch NP (2019) Science 365(6454):684 6. Ishizuka J, Sumita S, Daido A, Yanase Y (2019) Phys Rev Lett 123:217001 7. Yoshida T, Daido A, Kawakami N, Yanase Y (2019) Phys Rev B 99:235105

Curriculum Vitae

Shuntaro Sumita Condensed Matter Theory Laboratory RIKEN Cluster for Pioneering Research 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan E-mail: [email protected] Tel: +81-48-462-1111 (Ext. 3306) Website: https://shuntarosumita.github.io/

Employment Apr 2020–Present Apr 2017–Mar 2020

Postdoctoral researcher, Condensed Matter Theory Laboratory, RIKEN Cluster for Pioneering Reserach, Japan JSPS Research Fellowship for Young Scientists (DC1), Kyoto University

Education March 2020 March 2017 March 2015

Ph.D. from Department of Physics, Kyoto University M.S. from Department of Physics, Kyoto University B.S. from Faculty of Science, Kyoto University

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Sumita, Modern Classification Theory of Superconducting Gap Nodes, Springer Theses, https://doi.org/10.1007/978-981-33-4264-4

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Curriculum Vitae

Referred Papers 1. Sumita S, Yanase Y (2020) Superconductivity induced by fluctuations of momentum-based multipoles. Phys Rev Res 2:033225 2. Ishizuka J, Sumita S, Daido A, Yanase Y (2019) Insulator-metal transition and topological superconductivity in UTe2 from a first-principles calculation. Phys Rev Lett 123:217001 3. Sumita S, Nomoto T, Shiozaki K, Yanase Y (2019) Classification of topological crystalline superconducting nodes on high-symmetry lines: point nodes, line nodes, and Bogoliubov Fermi surfaces. Phys Rev B 99:134513 4. Sumita S, Yanase Y (2018) Unconventional superconducting gap structure protected by space group symmetry. Phys Rev B 97:134512 5. Kobayashi S, Sumita S, Yanase Y, Sato M (2018) Symmetry-protected line nodes and Majorana flat bands in nodal crystalline superconductors. Phys Rev B 97:180504(R) 6. Sumita S, Nomoto T, Yanase Y (2017) Multipole superconductivity in nonsymmorphic Sr2 IrO4 . Phys Rev Lett 119:027001 7. Sumita S, Yanase Y (2016) Superconductivity in magnetic multipole states. Phys Rev B 93:224507