Non-Bloch Band Theory of Non-Hermitian Systems (Springer Theses) 9811918570, 9789811918575

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Springer Theses Recognizing Outstanding Ph.D. Research

Kazuki Yokomizo

Non-Bloch Band Theory of Non-Hermitian Systems

Springer Theses Recognizing Outstanding Ph.D. Research

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Kazuki Yokomizo

Non-Bloch Band Theory of Non-Hermitian Systems Doctoral Thesis accepted by Tokyo Institute of Technology, Tokyo, Japan

Author Dr. Kazuki Yokomizo Condensed Matter Theory Laboratory RIKEN Wako, Saitama, Japan

Supervisor Prof. Shuichi Murakami Department of Physics Tokyo Institute of Technology Tokyo, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-19-1857-5 ISBN 978-981-19-1858-2 (eBook) https://doi.org/10.1007/978-981-19-1858-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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

Supervisor’s Foreword

Recent studies on non-Hermitian systems have revealed rich varieties of their intriguing physical properties. They are not mere generalization of Hermitian systems, but have their own unique features. For example, an interplay between symmetry and topology is quite different between Hermitian and non-Hermitian systems. Because various theoretical tools developed in Hermitian systems cannot be used in non-Hermitian systems, it is a challenging task to theoretically understand physical properties of non-Hermitian systems. This book is concerned with formulation of the non-Bloch band theory for nonHermitian systems, which is non-Hermitian extension of the Bloch band theory. The Bloch band theory is a fundamental theory in condensed matter physics, describing nature of eigenmodes of waves in crystals. This book, based on the dissertation of Dr. Kazuki Yokomizo, theoretically establishes the non-Bloch band theory in one-dimensional non-Hermitian systems, describing an asymptotic eigenspectrum in the limit of a large system size under open boundary conditions. This theory is nontrivial in itself because translational symmetry is absent under open boundary conditions. This theory is a great source of discoveries of beautiful, simple, and nontrivial properties of non-Hermitian systems. In particular, while people have believed that the bulk-edge correspondence in topological phases is violated in nonHermitian systems, it is shown to hold true through the non-Bloch band theory. Thus, the non-Bloch band theory established in this book is a fundamental theory for non-Hermitian systems, widely applicable to a wide range of systems. Tokyo, Japan December 2021

Prof. Shuichi Murakami

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Parts of this thesis have been published in the following journal articles: 1.

2.

3.

4.

Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch band theory in bosonic Bogoliubov-de Gennes systems”, Phys. Rev. B 103, 165123 (2021). Kazuki Yokomizo and Shuichi Murakami, “Topological semimetal phase with exceptional points in one-dimensional nonHermitian systems”, Phys. Rev. Research 2, 043045 (2020). Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch band theory and bulk-edge correspondence in non-Hermitian systems”, Prog. Theor. Exp. Phys. 2020, 12A102 (2020). Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch Band Theory of Non-Hermitian Systems”, Phys. Rev. Lett. 123, 066404 (2019).

vii

Acknowledgements

I would like to express my deepest appreciation goes to Prof. Shuichi Murakami. He gave comments and suggestions of inestimable value throughout the course of my study. In particular, his insight for previous researches paved a way to accomplish this work. Without his support and persistent help, this thesis would not have been completed. I am also particularly grateful for being able to travel and to participate in many conferences. I would also like to appreciate colleagues, faculties, and secretaries in the theoretical condensed matter physics groups in the Department of Physics at Tokyo Institute of Technology. I would especially like to be thankful to Ryo Takahashi, Ken Osumi, and Yutaro Tanaka. They gave me very useful comments for my researches and supported me. Outside Tokyo Institute of Technology, I am very grateful to Prof. Zhong Wang and Dr. Ryo Okugawa. Prof. Zhong Wang gave insightful comments and suggestions for the non-Bloch band theory and took care of me when I visited Tsinghua University. Dr. Ryo Okugawa discussed the non-Hermitian skin effect with me and gave me objective comments about my researches. Discussion with them have been illuminating a way to further development of my work. Finally, I would also like to acknowledge financial support from the Japan Society for the Promotion of Science and CREST, Japan Science and Technology Agency.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4

2 Hermitian Systems and Non-Hermitian Systems . . . . . . . . . . . . . . . . . . . 2.1 Topological Phases in Hermitian Systems . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Topological Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Topological Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Non-Hermitian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Parity-Time Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Exceptional Point, Exceptional Ring, and Exceptional Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Violation of the Bulk-Edge Correspondence in a Non-Hermitian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 9 12 14 16 16

3 Non-Hermitian Open Chain and Periodic Chain . . . . . . . . . . . . . . . . . . . 3.1 Generalized Brillouin Zone and Energy Spectrum in Open Chain . . . 3.2 Energy Spectrum in Periodic Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Comparison Between Open Chain and Periodic Chain . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 32 32 33 33

4 Non-Bloch Band Theory of Non-Hermitian Systems and Bulk-Edge Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Non-Bloch Band Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Condition for the Generalized Brillouin Zone . . . . . . . . . . . . . 4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 35 37 39

18 20 25

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Contents

4.2.1 Non-Hermitian Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . 4.2.2 Ladder Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Ladder Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Bulk-Edge Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Phase Diagram in the Non-Hermitian Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Topological Semimetal Phase with Exceptional Points in One-dimensional Non-Hermitian Systems . . . . . . . . . . . . . . . . . . . . . . . 5.1 Non-Hermitian Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . . . . . . . 5.1.1 Reinvestigation of the Energy Spectra . . . . . . . . . . . . . . . . . . . . 5.1.2 Mechanism for the Appearance of the Gapless Phase . . . . . . . 5.1.3 Winding Number, Exceptional Points, and Topological Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 TRS-Unbroken Region, STS-Unbroken Region, and TRS/STS-Broken Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Topological Semimetal Phase with Exceptional Points . . . . . . . . . . . . 5.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Winding Number and Creation and Annihilation of the Exceptional Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Symmetry-Breaking Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Instability of the Gapless Phase in a 1D Non-Hermitian System with Pseudo-Particle-Hole Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Pseudo-Particle-Hole Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Z2 Topological Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Generalized Non-Hermitian Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bosonic Bogoliubov-de Gennes Systems . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Real-Space Bogoliubov-de Gennes Hamiltonian . . . . . . . . . . . 6.1.2 Non-Bloch Band Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bosonic Kitaev-Majorana Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Non-Hermitian Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Analytical Representation of the Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 45 47 48 48 52 54 55 55 57 57 57 58 60 61 62 63 64 65 66 66 66 69 71 71 73 73 73 75 77 77 83 84 84

Contents

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7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 1

Introduction

1.1 Background The conventional Bloch band theory in crystals is fundamental for describing electronic structure. By introducing the Bloch wave vector k, energy bands calculated within a unit cell reproduce energy levels of a large crystal with open boundaries. Here it is implicitly assumed that in the limit of a large system size, electronic states are almost equivalent between a system with open boundaries and one with periodic boundaries, represented by the Bloch wave function with real k. This is because the electronic states extend over the system in both cases, implying that the effects of boundary conditions become negligible in the limit of a large system size. Since the discovery of the quantum Hall effect, topological phases have been one of the most studied themes in condensed matter physics. In many previous studies, rich aspects of the topological phases have been theoretically investigated, and topological phenomena have been experimentally observed. For example, in crystals, as a manifestation of topology, there appears bulk-edge correspondence between a topological invariant defined in terms of the Bloch wave functions and the existence of topological edge states localized at the ends of the system. In fact, the bulk-edge correspondence is realized in some materials, and this result justifies the description of the Bloch band theory in topological systems. Recently, the notion of topology has been extended from electronic systems to artificial systems, such as optical-lattice systems and metamaterials. In recent years, interest in studies on non-Hermitian physics has been rapidly growing, both in theories and in experiments. Historically, in a scattering problem, the resonant scattering in terms of a complex potential was the first study on nonHermitian physics [1]. After that, Ref. [2] proposed that Anderson localization, such as flux pinning in a superconductor, can be studied by using a non-Hermitian model. Recently, with the development of laser technology, non-Hermitian systems can be experimentally realized in various realistic systems, and studies on non-Hermitian physics have been dramatically progressing. Such systems are described by a non-

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_1

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1 Introduction

Hermitian Hamiltonian, and it is useful for studying nonequilibrium systems, which exchange energies and particles with an external environment. Non-Hermitian systems emerge in various fields of classical physics and quantum physics. In classical systems, such as a spring-mass system, an optical system, and an electric-circuit system gain and loss lead to non-Hermitian terms in eigenvalue equations. In particular, a parity-time symmetry [3] has been intensively studied to realize some phenomena unique to non-Hermitian systems, particularly in many optical systems [4–6], as explained in detail in Chap. 2. Besides, topological effects have been investigated in optical cavities which can be understood as a semiclassical analogy of nonHermitian quantum systems [7]. On the other hand, an open quantum system, such as a cold-atom system, is one of the most useful platforms realizing non-Hermitian systems [8]. For example, Ref. [9] theoretically proposed the experimental realization of asymmetric hopping amplitude in a one-dimensional (1D) optical lattice, where the system has the hopping amplitude to right different from that to left. Additionally, some previous work studied topological aspects in cold-atom systems with Bose-Einstein condensation [10–12]. Because of topological edge states, the systems exhibit remarkable phenomena induced by non-Hermiticity. We note that such systems can be described by the Bogoliubov-de Gennes (BdG) Hamiltonian. Besides, one of the origins of non-Hermiticity is a many-body correlation effect, for example, in an electron system [13]. Among many studies on non-Hermitian systems, one of the most intriguing topics is how non-Hermitian effects affect topological physics. Reference [14] first proposed that a topological invariant defined in a 1D non-Hermitian tight-binding model takes a fractional value, and localized edge states appear corresponding to the nonzero topological invariant. The previous work also mentioned the generalization of the bulk-edge correspondence in non-Hermitian systems. After this proposal, the bulk-edge correspondence in non-Hermitian systems has been much studied in theories. However, it has been controversial in this field because many previous works have revealed the violation of the conventional bulk-edge correspondence in nonHermitian systems. Hence, while the description of eigenstates in terms of the real wave vector k is justified in Hermitian systems by the conventional Bloch band theory, it is not the case in non-Hermitian systems. With this background, pioneering previous research [15] proposed a non-Hermitian skin effect. The non-Hermitian skin effect means that bulk eigenstates of a non-Hermitian system are localized at either end of an open chain. This phenomenon causes a difference between energy eigenvalues under an open boundary condition and those under a periodic boundary condition. Furthermore, the previous work also mentioned that the eigenstates behave as “non-Bloch” waves, where the wave functions are well approximated by Blochlike waves with a complex wave number. We will discuss these points in Chaps. 2 and 3. Since then, the non-Hermitian skin effect has been one of the hottest topics in researches on non-Hermitian physics. In particular, many previous works have been focusing on the topological structure of the non-Hermitian skin effect. Interestingly, by defining a topological invariant, called energy winding number, Refs. [16] and [17] proposed a new type of bulk-edge correspondence in non-Hermitian systems.

1.1 Background

3

Namely, when the energy winding number in a periodic chain takes nonzero values, bulk eigenstates are localized at the edges of an open chain. Thus the correspondence between a non-Hermitian open chain and a non-Hermitian periodic chain was revealed. On the other hand, a lack of research on a non-Hermitian open chain has been a long-standing issue. Although Ref. [15] showed the calculation of energy bands and Brillouin zone in 1D non-Hermitian systems, its method is applicable only to a very limited class of non-Hermitian systems, and it is needed to extend this method to general cases. Thus, due to the non-Hermitian skin effect, we should modify the Bloch band theory in non-Hermitian systems in order to discuss the bulk-edge correspondence and to study non-Hermitian physics. Based on the idea of non-Hermitian skin effect and the description of non-Bloch wave, in this thesis, we construct a non-Bloch band theory of non-Hermitian crystalline systems. From our theory, in the non-Hermitian systems, we can determine the Brillouin zone reproducing energy bands in a long open chain. Thus the non-Bloch band theory is fundamental for non-Hermitian physics, and therefore, this theory can reveal various aspects of the non-Hermitian systems. Firstly, a byproduct of our theory is that one can prove the bulk-edge correspondence between a topological invariant defined in the bulk and the existence of localized edge states in general non-Hermitian systems. In fact, we will demonstrate the bulk-edge correspondence in the non-Hermitian Su-Schrieffer-Heeger (SSH) model in Chap. 4. Secondly, we can find a new type of a gapless phase in non-Hermitian systems in terms of the non-Bloch band theory. We will show that this gapless phase is stabilized because of the modification of the Brillouin zone in Chap. 5. Thirdly, we can study the nonHermitian nature of a bosonic system described by a Bogoliubov-de Gennes (BdG) Hamiltonian. While some previous works studied non-Hermiticity in a finite open chain by numerical calculation, non-Hermitian physics in a bosonic BdG system in the thermodynamic limit has been unrevealed yet. Hence the non-Bloch band theory can clarify non-Hermitian properties in this system. In fact, we will show rich aspects of the non-Hermitian skin effect, i.e., infinitesimal instability and reentrant behavior, in Chap. 6.

1.2 Outline This thesis is organized to establish the non-Bloch band theory and to study various properties in the non-Hermitian systems as mentioned in Sect. 1.1. The organization of this thesis is the following. In Chap. 2, first of all, we start with the review of topological physics in Hermitian systems. We introduce two kinds of topological insulators and explain how these insulating phases can be topologically classified. Furthermore, in the SSH model, we derive a Z topological invariant from a Q matrix and demonstrate the bulk-edge correspondence. Next, we explain non-Hermitian systems. We particularly focus on a parity-time symmetry, which is one of the most studied topics in non-Hermitian systems. Throughout the discussion, we give the notion of an exceptional point

4

1 Introduction

and explain that such non-Hermitian degeneracy can appear not only in parameter space but also in momentum space. Finally, we show the violation of the bulk-edge correspondence in the non-Hermitian SSH model, caused by the non-Hermitian skin effect. In Chap. 3, in order to discuss the non-Hermitian skin effect, we compare a non-Hermitian system with an open boundary condition with that with a periodic boundary condition. In particular, we deeply analyze the difference between the energy eigenvalues and the behavior of the eigenstates by using a simple model. In Chap. 4, we establish the non-Bloch band theory in a tight-binding nonHermitian system. We show a way to determine the generalized Brillouin zone (GBZ) β = eik for the complex Bloch wave number k ∈ C. Then we see some aspects of the GBZ in some models. Furthermore, in the non-Hermitian SSH model, we demonstrate the bulk-edge correspondence between the topological invariant defined by the GBZ and the appearance of the topological edge states. Finally, we comment on the existence of the gapless phase in the non-Hermitian SSH model in this chapter. In Chap. 5, according to the discussion in Chap. 4, we investigate the gapless phase which appears in the non-Hermitian SSH model. Then we show that in 1D non-Hermitian systems with a sublattice symmetry and a time-reversal symmetry, a gapless phase with exceptional points is stabilized because of the unique features of the GBZ. Furthermore, we also find that each energy band is divided into three regions depending on the symmetry of the eigenstates. In Chap. 6, we study the non-Bloch band theory in bosonic BdG systems. In terms of our theory, we can calculate the GBZ and can study non-Hermitian properties in terms of the GBZ. As an example, by investigating the bosonic Kitaev-Majorana chain, we show rich aspects of the non-Hermitian skin effect, such as infinitesimal instability and reentrant behavior. In Chap. 7, we summarize this thesis and comment on the outlook of the non-Bloch band theory.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Gamow G (1928) Z Phys A 51:204 Hatano N, Nelson DR (1996) Phys Rev Lett 77:570 Bender CM, Boettcher S (1998) Phys Rev Lett 80:5243 Feng L, El-Ganainy R, Ge L (2017) Nat Photonics 11:752 El-Ganainy R, Makris KG, Khajavikhan M, Musslimani ZH, Rotter S, Christodoulides DN (2018) Nat Phys 14:11 Özdemir SK, ¸ Rotter S, Nori F, Yang L (2019) Nat Mater 18:783 Ota Y, Takata K, Ozawa T, Amo A, Jia Z, Kante B, Notomi M, Arakawa Y, Iwamoto S (2020) Nanophotonics 9:547 Ashida Y, Gong Z, Ueda M (2020) Adv Phys 69:249 Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M (2018) Phys Rev X 8:031079 Engelhardt G, Brandes T (2015) Phys Rev A 91:053621 Galilo B, Lee DKK, Barnett R (2015) Phys Rev Lett 115:245302

References 12. 13. 14. 15. 16. 17.

Engelhardt G, Benito M, Platero G, Brandes T (2016) Phys Rev Lett 117:045302 Kozii V, Fu L arXiv:1708.05841 Lee TE (2016) Phys Rev Lett 116:133903 Yao S, Wang Z (2018) Phys Rev Lett 121:086803 Okuma N, Kawabata K, Shiozaki K, Sato M (2020) Phys Rev Lett 124:086801 Zhang K, Yang Z, Fang C (2020) Phys Rev Lett 125:126402

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Chapter 2

Hermitian Systems and Non-Hermitian Systems

2.1 Topological Phases in Hermitian Systems In this section, we review the topological physics in Hermitian systems. First, we explain the relationship between physics and topology throughout the discovery of a topological phenomenon. Next, we introduce the topological periodic table obtained by a topological classification. Finally, in the SSH model, we demonstrate the bulkedge correspondence expected from a topological classification.

2.1.1 Quantum Hall Effect The quantum Hall effect is the first phenomenon described by topological band theory in condensed matter physics [1–3]. When a magnetic field is added to a twodimensional (2D) electron gas, the Hall conductivity σx y is quantized as an integer multiple of e2 / h, explicitly written as σx y = N

e2 , (N = 0, ±1, ±2, . . . ) . h

(2.1)

This phenomenon is called quantum Hall effect, and the quantization of the Hall conductivity can be described by the linear response theory. In this subsection, we investigate such a 2D electronic system in the x y plane without a time-reversal symmetry (TRS). To study the Hall conductivity, we calculate a transverse current response when an electric field is applied perpendicular to the magnetic field. Then the Hall conductivity can be written in terms of the Kubo formula [4] as

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_2

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8

2 Hermitian Systems and Non-Hermitian Systems





⎤ ∂Hk ∂Hk u u u u m,k n,k ⎥   ⎢ n,k ∂k x m,k ∂k y ie2    ⎢ =− 2 f E n,k − c.c.⎥   2 ⎣ ⎦, hL E n,k − E m,k n m=n k ⎡

σx y

(2.2)   where f E n,k is the Fermi distribution function, L 2 is the system size, |u n,k  is the periodic part of the Bloch wave function in the nth band with the Bloch wave vector k, Hk is the Bloch Hamiltonian, and E nk is the energy eigenvalue of the nth band. At zero temperature, by using Eq. (2.66), Eq. (2.2) can be rewritten as σx y = −

e2 h



νn ,

(2.3)

d2k Bn,z (k) , 2π

(2.4)

n:occupied

where νn is defined over the Brillouin zone as  νn ≡ B.Z.

called Chern number. Furthermore, Bn,z (k) is the z component of the Berry curvature defined as

 ∂u n,k ∂u n,k × . (2.5) B n (k) = i ∂k ∂k In the following, we show that the Chern number νn is always an integer. The Berry curvature and the Berry connection in the momentum representation are given in Eq. (2.67). Therefore, if we naively apply the Stokes’ theorem to Eq. (2.4), we can rewrite this formula as  dk · An (k) , (2.6) νn = 2π ∂B.Z. where ∂B.Z. is the boundary of the 2D Brillouin zone [Fig. 2.1a]. The integral always vanishes, and νn is zero due to the periodicity of the Brillouin zone. This conclusion is valid only when the Bloch wave function is continuous in the whole Brillouin zone. On the other hand, when the Bloch wave function has a singularity on the Brillouin zone, νn may take nonzero values. Here we take gauge I on one region so that the Bloch wave function has no singularity, and we take a gauge II on the other region which includes a singularity of the Bloch wave function [Fig. 2.1b]. Let C denote the boundary between the regions I and II. Then Eq. (2.6) can be rewritten in terms of the corresponding Berry connections AIn (k) and AIIn (k) as  νn = C

 d k  II · An (k) − AIn (k) . 2π

(2.7)

2.1 Topological Phases in Hermitian Systems

(a)

9

(b)

ky

ky I

kx

kx II

Brillouin Zone

C

Fig. 2.1 Regions for the integral of the Berry curvature on the Brillouin zone. a The Bloch wave function has no singularities in the Brillouin zone. b The Bloch wave function has no singularities in region I, but it has a singularity in region II. C is the boundary between the regions I and II

Since the two Bloch wave functions |u In,k  and |u IIn,k  on the two regions are related to each other by the gauge transformation |u In,k  = |u IIn,k  eiχn (k) , we obtain νn =

1 2π

 d k · ∇ k χn (k) .

(2.8)

C

The integral included in Eq. (2.8) is always an integer multiple of 2π because of the single-valuedness of the wave function in each region. Therefore, we conclude that the Chern number (2.4) is always an integer. As a result, the Hall conductivity (2.3) is quantized as an integer multiple of e2 / h.

2.1.2 Topological Insulator Since the Chern number (2.4) is odd under time reversal, it always vanishes in systems with the TRS. Thus a superposition of two quantum Hall systems with opposite signs of the Chern number, which recovers the TRS, is a trivial insulator in terms of the Chern number. However, it has been found that it can be topologically nontrivial in terms of a Z2 topological invariant. Moreover, such a superposition of two quantum Hall systems with opposite spins can be naturally realized by spin-orbit coupling. This system is called topological insulator (TI) which shows a quantum spin Hall effect. They have been proposed as a special case of a spin Hall effect for spinful particles [5–7]. After the theoretical proposal, some TIs have been experimentally observed in the HgTe/CdTe quantum well as a 2D TI [8] and in Bi2 Se3 and Bi2 Te3 as a three-dimensional (3D) TI [9–11]. In the following, we review topological invariants characterizing the TI phase.

10

2 Hermitian Systems and Non-Hermitian Systems

First of all, we define the Z2 topological invariant characterizing a 2D TI. In a spinful system with the TRS, the time-reversal operator T can be written as T = iσ y K , where σ y is the y component of the Pauli matrices and K is the complexconjugate operator. We note that T satisfies T 2 = −1. Since the system is invariant under the time-reversal operation, the Hamiltonian of this system satisfies [H, T ] = 0. Then, for the Bloch Hamiltonian Hk = e−i k·r H ei k·r , we can get T Hk T −1 = H−k .

(2.9)

For the Bloch eigenstate |u n,k , if k = −k, one can immediately find that |u n,k  and T |u n,k  have the same energy eigenvalue at the same wave vector. This is called Kramers’ degeneracy. In the momentum space, there can exist wave vectors for which k and −k are equivalent because of the periodicity of the Brillouin zone. It can be explicitly written as k ≡ −k, (mod G) , (2.10) where G are reciprocal-lattice vectors. Such wave vector is called time-reversalinvariant momentum (TRIM). For example, there are four TRIM in the 2D Brillouin zone given by 1 (2.11)  i = (n 1 G 1 + n 2 G 2 ) , (n 1 , n 2 = 0, 1) , 2 where G 1 and G 2 are primitive vectors in the reciprocal lattice. Here let  i (i = 1, . . . , 4) denote the four TRIM corresponding to k = 0, G 1 /2, G 2 /2, (G 1 + G 2 ) /2 as shown in Fig. 2.2a. As shown below, the Z2 topological invariant is defined in terms of the Bloch eigenstates at these four TRIM. To define the Z2 topological invariant, we use the theory of time-reversal polarization [12]. In terms of the occupied Bloch eigenstate, we define a U (2N ) matrix as (a)

(b)

(001)

(01)

Γ5 Γ6

Γ3

Γ7 Γ8

Γ4

Γ1

Γ3

(100) Γ1

Γ2

(10)

(010) Γ2

Γ4

Fig. 2.2 Schematic figures of the time-reversal-invariant-momenta (TRIM). a There are four TRIM in the two-dimensional Brillouin zone. b There are eight TRIM in the three-dimensional Brillouin zone

2.1 Topological Phases in Hermitian Systems

11

wmn (k) = u m,−k |T |u n,k , (m, n = 1, . . . , 2N ) ,

(2.12)

where 2N is the number of occupied bands. For simplicity, we assume that there are no degeneracies except for the Kramers’  degeneracy. Since each Bloch eigenstate forms a Kramers’ pair T |u n,k , |u n,−k  , the matrix wmn (k) becomes an antisymmetric matrix. Then, this matrix can be characterized by the Pfaffian, whose square is equal to the determinant. Hence, we define an index δi at each TRIM as Pf (w ( i )) δi ≡ √ . det (w ( i ))

(2.13)

Then, in a 2D system with the TRS, the Z2 topological invariant ν can be defined as (−1)ν =

4 

δi .

(2.14)

i=1

One can find that ν becomes either 0 or 1 because the indices δi (i = 1, . . . , 4) only take the values ±1. It has been shown that if the system has ν = 1, it becomes a TI, showing the quantum spin Hall effect. We note that the indices δi (i = 1, . . . , 4) depend on√a gauge degree of freedom but their product is gauge invariant, and the branch of det w ( i ) should be chosen so that it is continuous over the half of the Brillouin zone. The choice of the branch is needed in calculating the Bloch wave functions, in order to eliminate ambiguity in evaluating Eq. (2.13). Next, similar to a 2D system, we can also characterize the TI phases in a 3D system by using the same index δi . In this case, there are eight TRIM in the 3D Brillouin zone given by i =

1 (n 1 G 1 + n 2 G 2 + n 3 G 3 ) , (n 1 , n 2 , n 3 = 0, 1) , 2

(2.15)

where G 1 , G 2 , and G 3 are primitive vectors in the reciprocal lattice. We show the eight TRIM in Fig. 2.2b. Then, this system is characterized by four Z2 topological invariants ν0 ; (ν1 ν2 ν3 ) defined as ⎧ 8 ⎪  ⎪ ν0 ⎪ ⎪ = δi , (−1) ⎪ ⎨ i=1 (2.16)  ⎪ ⎪ ⎪ (−1)ν j = δ , j = 1, 2, 3) . ( ⎪ i ⎪ ⎩ i=(n 1 n 2 n 3 );n j =1

The system with ν0 = 1 is called strong TI regardless of the values of the indices νi (i = 1, 2, 3). On the other hand, when ν0 = 0 and one of the indices νi (i = 1, 2, 3) takes a nonzero value, the system is called weak TI. While the weak TI can be regarded as layered 2D TIs stacked along the direction represented by the Miller indices (ν1 ν2 ν3 ), three-dimensionality plays an essential role in the strong TI.

12

2 Hermitian Systems and Non-Hermitian Systems

2.1.3 Topological Classification In terms of the Altland-Zirnbauer (AZ) symmetry classes, the quantum Hall insulator and the TI belong to class A and class AII, respectively. The topological classifications of a 2D system in class A and 2D and 3D systems in class AII are given by Z and by Z2 , respectively. These topological classifications are consistent with the results in Sects. 2.1.1 and 2.1.2. In the following, we review the AZ symmetry classes [13, 14] and the method of the topological classification in a fermionic single-particle system [15, 16]. 2.1.3.1

Altland-Zirnbauer Symmetry Classes

For a free-fermion system described by the second-quantized real-space Hamiltonian  H= Hi j ci† c j , (2.17) i, j

where ci is an annihilation operator of a fermion at the ith site, and Hi j is a Hermitian matrix satisfying H ji∗ = Hi j ; in general, there are three fundamental internal symmetries, such as a time-reversal symmetry (TRS), a particle-hole symmetry (PHS), and a chiral symmetry (CS). The time-reversal operator T and the particle-hole operator C are antiunitary operators, and these are written as T = UT K , C = UC K ,

(2.18)

respectively, where K is the complex-conjugate operator, and UT and UC are unitary matrices. Depending on the corresponding system, these operators satisfy either T 2 = +1 or T 2 = −1, and either C 2 = +1 or C 2 = −1. For example, the timereversal operator given in Sect. 2.1.2 satisfies T 2 = −1. Furthermore, the chiral operator  is given as the product of two operators T and C, i.e.,  = T C. Hence,  is a unitary matrix, and it satisfies  2 = +1. When the system is an invariant under such operations, the Hamiltonian (2.17) satisfies T H T −1 = H, C H C −1 = −H,  H  −1 = −H,

(2.19)

respectively. Now, three internal symmetries label ten internal symmetry classes, leading to the AZ symmetry classes. The AZ symmetry classes are summarized in Table 2.1. 2.1.3.2

Topological Periodic Table

We explain the topological classification in terms of the AZ symmetry classes. In the following, we study an insulating system in which the Fermi energy lies between the valence bands and the conduction bands. Here we assume that the valence bands and the conduction bands consist of M and N energy levels, respectively. In the

2.1 Topological Phases in Hermitian Systems

13

Table 2.1 Altland-Zirnbauer symmetry classes. The numbers 0 and 1 represent whether the Hamiltonian (2.17) satisfies Eq. (2.19) or not, and the numbers ±1 express the values of T 2 and C 2 Symmetry class Time-reversal operator T Particle-hole operator C Chiral operator  A AI AII AIII BDI CII D C DIII CI

0 +1 −1 0 +1 −1 0 0 −1 +1

0 0 0 0 +1 −1 +1 −1 +1 −1

0 0 0 1 1 1 0 0 1 1

topological classification of such a system, the energy eigenvalues of the Hamiltonian (2.17) do not matter. Instead, we flatten the energy eigenvalues to ±1 in order to classify the insulating phase in this system. To this end, by using the eigenstates of the Bloch Hamiltonian Hk = e−i k·r H ei k·r , we define the Q operator as Q (k) =

N 

+ |u + n,k u n,k | −

n=1

M 

− |u − n,k u n,k |,

(2.20)

n=1

− where the energy eigenstates |u i,k  (i = 1, . . . , M) and |u +j,k  ( j = 1, . . . , N ) belong to the valence bands and conduction bands, respectively. The Q operator can be diagonalized by the (N + M) × (N + M) unitary matrix   + + + − − − − U (k) = |u + (2.21) 1,k u 1,k |, . . . , |u N ,k u N ,k |, |u 1,k u 1,k |, . . . , |u M,k u M,k |

into U (k) Q (k) U

−1

 (k) = , =

1l N O O −1l M

 ,

(2.22)

where 1l N and 1l M are the N × N and M × M identity matrices, respectively. We note that Eq. (2.22) is a matrix representation of the Q operator. Furthermore, Eq. (2.22) tells us that the Mth eigenvalues in the valence bands are degenerate at −1, and the N th eigenvalues in the conduction bands are degenerate at +1. Thus the energy eigenvalues of the Hamiltonian (2.17) are flattened to ±1 by the unitary matrix (2.21). Now the system is deformed to a topologically trivial insulating system by the flattening (2.22). On the other hand, if there exists topologically different Q matrix other than the diagonal matrix given in Eq. (2.22), we can get a topologically nontrivial insulating phase. Here, by using the homotopy theory, we can determine whether the system has a topologically nontrivial phase or not. Omitting the mathematical details, in a d-dimensional system, the topological classification for each AZ symmetry class is given in Table 2.2.

14

2 Hermitian Systems and Non-Hermitian Systems

Table 2.2 Topological periodic table. The number 0 means that a system has only a topologically trivial insulating phase Symmetry class T C  d=0 1 2 3 4 A AIII AI BDI D DIII AII CII C CI

0 0 +1 +1 0 −1 −1 −1 0 +1

2.1.3.3

0 0 0 +1 +1 +1 0 −1 −1 −1

0 1 0 1 0 1 0 1 0 1

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

Z 0 2Z 0 0 0 Z Z2 Z2 0

Class AIII

When the system belongs to class AIII, we can get a simple result for a topological invariant. Due to the CS, the valence bands and the conduction bands of this system are symmetrical to each other with respect to E = 0; therefore, N = M. Then, with a diagonal matrix representation for , we can write the Q matrix in an off-diagonal form as   O q (k) , (2.23) Q (k) = q † (k) O where q (k) is an N × N unitary matrix. Table 2.2 tells us that in this case, the topologically nontrivial insulating phase is realized in (2n + 1)-dimensional systems. Then, we can define the topological invariant νw by using q (k) as [16]  νw =

d 2n+1 k

(−1)n n! (2n + 1)!



    i n+1 μ1 ···μ2n+1  †

Tr q (k) ∂μ1 q (k) · · · q † (k) ∂μ2n+1 q (k) , 2π

(2.24) where

is the totally antisymmetric tensor. In the next subsection, we will discuss the relation between this topological invariant and the appearance of the edge states in the SSH model. μ1 ···μ2n+1

2.1.4 Su-Schrieffer-Heeger Model In this subsection, we review the topological aspects of the SSH model [17]. This system is a one-dimensional (1D) Hermitian system, and the real-space Hamiltonian of this system is given by H=

 n

 † † t1 cn,A cn,B + t2 cn+1,A cn,B + h.c. ,

(2.25)

2.1 Topological Phases in Hermitian Systems

15

where the parameters t1 and t2 are real. By the Fourier transformation, we can get the Bloch Hamiltonian as   0 t1 + t2 e−ik , (2.26) H (k) = t1 + t2 eik 0 where k is the Bloch wave number. Then the energy eigenvalues can be explicitly written as  E ± (k) = ± t12 + t22 + 2t1 t2 cos k. (2.27) Since the Bloch Hamiltonian has the chiral symmetry expressed as  H (k)  −1 = −H (k) ,  =



1 0 0 −1

 (2.28)

and the time-reversal symmetry expressed as T H (k) T

−1

 = H (k) , T =

01 10

 K,

(2.29)

where K is the complex-conjugate operator, this system belongs to class BDI, and it can be topologically classified in terms of the Z topological invariant as shown in Table 2.2. Now we define the Z topological invariant in terms of the Q matrix. From Eqs. (2.26) and (2.27), the Q matrix can be written as H (k) Q (k) = t1 + t2 eik   0 q (k) , ≡ q −1 (k) 0

(2.30)

and the topological invariant νw , called winding number, can be defined as νw =

i 2π



dq q −1 (k) ,

(2.31)

C

where C is a closed path drawn by the function q (k) when k changes from −π to π . We note that Eq. (2.31) is the case of N = 1 and n = 0 in Eq. (2.24). Now Eq. (2.31) can be calculated as  0 if |t1 | > |t2 | , (2.32) νw = 1 if |t1 | < |t2 | . Thus Eq. (2.32) tells us that while this system is topologically trivial when |t1 | > |t2 |, it becomes a topological insulator when |t1 | < |t2 |. Importantly, when |t1 | < |t2 |, there exist edge states localized at both ends of a finite open chain as a manifestation

16

3

E

Fig. 2.3 Energy levels in the Su-Schrieffer-Heeger model with the finite system size L = 50 under an open boundary condition. The parameter t2 is set to be 1. We show the topological edge states in red

2 Hermitian Systems and Non-Hermitian Systems

0

-3 -3

-1

0

1

3

t1 of the nontrivial topology in the bulk. In fact, by calculating the energy levels under an open boundary condition, one can see the appearance of the topological edge states in this system as shown in Fig. 2.3. Furthermore, from this figure, we can also confirm that the gap in the bulk closes around E = 0 when |t1 | = |t2 |. In fact, in this case, the energy eigenvalues satisfy E + (0) = E − (0) = 0 (or E + (π ) = E − (π ) = 0). In conclusion, the nontrivial topological invariant corresponds to the existence of the topological edge states, and the topological invariant changes its value only when the bulk band gap closes. This correspondence is called bulk-edge correspondence.

2.2 Non-Hermitian Systems In this section, we review previous studies on non-Hermitian systems. Among various previous works, we particularly focus on non-Hermitian degeneracies where some energy eigenvalues become degenerate and the corresponding eigenstates coalesce, which has been intensively studied in both theories and experiments. First of all, we explain the appearance of such degeneracies in parity-time (PT) symmetric systems and experimental realization of PT-symmetric systems. Next, we show that such non-Hermitian degeneracies can appear not only in parameter space but also in momentum space. Finally, we review the experimental realization of these nonHermitian degeneracies in a photonic system.

2.2.1 Parity-Time Symmetry Non-Hermitian systems are described by a non-Hermitian Hamiltonian. While the energy eigenvalues of a system in the conventional quantum mechanics are real,

2.2 Non-Hermitian Systems

17

such non-Hermitian systems have complex energy eigenvalues in general because the Hamiltonian is a non-Hermitian matrix. Nevertheless, the energy eigenvalues of non-Hermitian systems can take real values under some symmetries. Among them, a PT symmetry, which was first proposed in Ref. [18], has been intensively studied both in theories and experiments. When the Hamiltonian H satisfies [H, PT ] = 0,

(2.33)

where P is a parity operator, and T is a time-reversal operator, the system has the PT symmetry. We note that H is not invariant either under the parity operation or under the time-reversal operation. If the eigenstate |ψ of the Hamiltonian satisfies PT |ψ = eiθ |ψ (θ ∈ R), the PT symmetry is preserved. On the other hand, when |ψ does not satisfy this condition, the PT symmetry is spontaneously broken. In the following, we demonstrate the spontaneous breaking of the PT symmetry in a two-level model. The Hamiltonian of this model is written as   iγ α , (2.34) Htwo = α −iγ where the parameters α and γ take positive real values. This Hamiltonian preserves the PT symmetry because it satisfies (PT ) Htwo (PT )−1 = Htwo ,

(2.35)

where P is given by P = σx being the x-component of the Pauli matrices, and T is given by T = K being the complex-conjugate operator. Depending on the values of the parameters, the energy eigenvalues E ± of this model take either real values or pure imaginary values, explicitly represented as ⎧  ⎪ ± α 2 − γ 2 if α > γ , ⎪ ⎨ if α = γ , E± = 0 ⎪  ⎪ ⎩ ±i γ 2 − α 2 if α < γ .

(2.36)

Namely, while the energy eigenvalues become real when α > γ [Fig. 2.4a], they become pure imaginary when α < γ [Fig. 2.4c]. The system in the former case is in a PT-symmetry unbroken phase, and the system in the latter case is in a PT-symmetry broken phase. Importantly, at the boundary between the two phases, when α = γ , two energy eigenvalues E ± are degenerate [Fig. 2.4b]. In this case, the Hamiltonian is nondiagonalizable because the corresponding eigenstates coalesce. This is why this degeneracy is unique to non-Hermitian systems. In general, this degenerate point is called exceptional point [19]. Thus we can demonstrate the spontaneous breaking of the PT symmetry and the appearance of the exceptional point at the phase boundary.

18

2 Hermitian Systems and Non-Hermitian Systems Im(E)

(a) α > γ

Re(E)

Im(E)

(b) α = γ

Im(E)

(b) α < γ

Re(E)

Re(E)

Exceptional point

Fig. 2.4 Energy eigenvalues in the parity-time (PT) symmetric system, given in Eq. (2.34). a When α > γ , the energy eigenvalues become real, and the system is in a PT-symmetry unbroken phase. b When α = γ , the energy eigenvalues are degenerate, and the system has an exceptional point. c When α < γ , the energy eigenvalues become pure imaginary, and the system is in a PT-symmetry broken phase

In many previous works, the phase transition associated with the PT symmetry and some phenomena by using the non-Hermitian degeneracy, such as nonreciprocal transport, has been intensively investigated and experimentally realized. In fact, the PT-unbroken phase and PT-broken phase were realized in various kinds of systems [20–24]. Furthermore, by using the coalescence of eigenstates at an exceptional point, the unidirectional transmission [25, 26] and the unidirectional retroreflection on a metasurface [27] were also experimentally observed. The topology of the PT-symmetric phase has been also attracting much attention. Starting with the experimental study on the topological structure of an exceptional point [28], both theoretical and experimental works investigated how the PT symmetry affects the bulk-edge correspondence and the topological edge states [29–43]. Additionally, in terms of the global Berry phase [44], it was theoretically proposed that the localized edge state appears at the boundary between a PT-unbroken system and a PT-broken system [45]. After this proposal, this edge state was experimentally observed [46].

2.2.2 Exceptional Point, Exceptional Ring, and Exceptional Surface In recent years, there are some previous studies on the appearance of the nonHermitian degeneracies in the momentum space, instead of the parameter space. Now we review the theory of the non-Hermitian degeneracies, following Ref. [47]. First of all, we focus on the non-Hermitian Bloch Hamiltonian of a two-level system, written as (2.37) H (k) = (a0 (k) + ia1 (k)) σ0 + (b0 (k) + i b1 (k)) · σ ,   where σ0 and σ = σx , σ y , σz are the 2 × 2 identity matrix and the Pauli matrices, respectively, and a j (k) and b j (k) = b j,x (k) , b j,y (k) , b j,z (k) ( j = 0, 1) are real

2.2 Non-Hermitian Systems

19

functions of the wave vector k. The energy eigenvalues are given by E ± (k) = a0 (k) + ia1 (k) ±



b0 (k) · b0 (k) − b1 (k) · b1 (k) + 2i b0 (k) · b1 (k). (2.38) Then we can get the conditions for degeneracy of the two eigenenergies as b0 (k) · b0 (k) − b1 (k) · b1 (k) = 0, b0 (k) · b1 (k) = 0.

(2.39)

These conditions can be satisfied in d-dimensional non-Hermitian systems when d ≥ 2. In particular, when d = 2, we expect that these conditions are satisfied at isolated points in the momentum space, and they are exceptional points. The energy dispersion has a linear dispersion around such an exceptional point. We note that the whole momentum space must have an even number of exceptional points because of the Nielsen-Ninomiya theorem [48]. On the other hand, when d = 3, the nonHermitian degeneracy forms a closed loop, called exceptional ring. By adding some symmetries, the forms of the non-Hermitian degeneracies change. For example, when the system has the PT symmetry represented as (PT ) H (k) (PT )−1 = H (k) , PT = σ0 K ,

(2.40)

where K is the complex-conjugate operator, the Hamiltonian (2.37) can be transformed into HPT (k), explicitly written as HPT (k) = a0 (k) σ0 + b0,x (k) σx + b0,z (k) σz + ib1,y (k) σ y ,

(2.41)

and then, the energy eigenvalues are given by E ± (k) = a0 (k) ±

 2 2 2 b0,x (k) + b0,z (k) − b1,y (k).

(2.42)

Hence we can get the condition that the two energy eigenvalues are degenerate as 2 2 2 b0,x (k) + b0,z (k) = b1,y (k) .

(2.43)

In contrast to Eq. (2.39), the number of conditions for degeneracy of the two eigenenergies is reduced by one in this case. This indicates that the exceptional points and the exceptional ring are deformed to a closed loop and a closed surface in 2D and 3D non-Hermitian systems, respectively. The closed surface is called exceptional surface. Thus we can change the formation of the non-Hermitian degeneracies by decreasing degrees of freedom. In conjunction with studies on such non-Hermitian degeneracies [47, 49–64], the realization of exceptional points, exceptional rings, and exceptional surfaces have been much investigated. Among many previous works, it was theoretically proposed that in a strongly correlated electron system, exceptional points may appear in the momentum space [65–68]. In such systems, we can get a non-Hermitian

20

2 Hermitian Systems and Non-Hermitian Systems

(a-1) d=2

(a-2) d=2

(b-1) d=3

(b-2) d=3

ky

ky kx

kx ky

kz

kx

ky

kz

kx

Fig. 2.5 Non-Hermitian degeneracies in a two-dimensional and b three-dimensional momentum spaces. a Exceptional points in a-1 are deformed to an exceptional ring in (a-2). b An exceptional ring in b-1 is deformed to an exceptional surface in (b-2)

Hamiltonian by incorporating the imaginary part of the self-energy representing the lifetime of a quasiparticle into a one-body Hamiltonian. Furthermore, in this case, an open-ended curve, called bulk-Fermi arc, along which the real parts of the energy eigenvalues are degenerate, appears so as to connect the exceptional points. Importantly, Ref. [65] proposed that it can be observed by the angle-resolved photoemission spectroscopy. Additionally, some non-Hermitian phenomena have been discussed in strongly correlated electron systems. We will comment on recent studies on non-Hermiticity in such systems in Chap. 7. While the bulk-Fermi arc has not been yet experimentally observed in strongly correlated electron systems, its concept can be extended to other physical systems. In fact, by taking over this idea, the “bulk-Fermi arc” was experimentally observed in a 2D photonic crystal [69]. This is an indirect verification of the realization of a pair of exception points. Thus, a photonic crystal is useful for the demonstration of some non-Hermitian degeneracies. In fact, an exceptional ring was also realized in a 2D photonic crystal [70] and a 3D photonic crystal [71].

2.3 Violation of the Bulk-Edge Correspondence in a Non-Hermitian System In recent years, interest in studies on how non-Hermitian effects affect topological physics has been rapidly growing. In particular, the violation of the conventional bulk-edge correspondence in non-Hermitian systems has been a long-standing issue, and the reasons for this violation have been under debate. In some previous works [72–90], one of the controversies is that in non-Hermitian systems, the Bloch wave number was treated as real, similar to Hermitian ones. In order to solve this issue, other previous works studied the methods to treat the Bloch wave number as complex [91–98]. Among these previous works, Ref. [91] pointed out that the violation of the bulk-edge correspondence is caused by the difference between the behavior of an open chain and that of a periodic chain, and it also proposed a non-Hermitian skin effect. In the following, we briefly review the non-Hermitian skin effect.

2.3 Violation of the Bulk-Edge Correspondence in a Non-Hermitian System

21

To demonstrate the violation of the bulk-edge correspondence, we investigate the non-Hermitian SSH model proposed in Ref. [91]. It is defined on a 1D lattice, and each unit cell consists of two sublattices A and B. The real-space Hamiltonian of this system is given by     γ † γ † † † cn,A cn,B + t2 cn+1,A cn,B cn,A + t2 cn,B t1 + cn,B + t1 − cn+1,A , 2 2 n (2.44) where t1 , t2 , and γ are real parameters, and cn,μ is an annihilation operator at the sublattice μ(= A, B) in the nth unit cell. When we set the Bloch wave number k to be real, which corresponds to imply a periodic boundary condition to the system, we can get the Bloch Hamiltonian of this system as H=

 ⎞ γ + t2 e−ik t1 + 2 ⎟ ⎜ H (k) = ⎝  ⎠, γ + t2 eik t1 − 0 2 ⎛

0

(2.45)

and then, the energy eigenvalues can be calculated as & E ± (k) = ± t12 + t22 + 2t1 t2 cos k + it2 γ sin k −

γ2 . 4

(2.46)

One can find from Eq. (2.46) that the bulk gap of this system closes when ⎧ γ ⎪ ⎨ −t2 ± and k = 0, 2 t1 = γ ⎪ ⎩ t2 ± and k = π. 2

(2.47)

As discussed in Sect. 2.1.4, the bulk gap closing described by Eq. (2.47) divides the parameter regions with and without the topological edge states. In order to confirm this, we show the numerical calculation of the energy levels in the non-Hermitian SSH model [Fig. 2.6]. In this figure, the green dashed lines express the values of the parameter t1 given by Eq. (2.47), and the black lines and the blue lines are the energy levels under an open boundary condition and those under a periodic boundary condition, respectively. One can see that the green dashed lines and the blue lines do not match the result for an open chain. Thus the bulk-edge correspondence between the bulk gap closing and the appearance of the topological edge states seems to be violated. From the above discussion, we conclude that in non-Hermitian systems, the energy eigenvalues in an open chain and those in a periodic chain are different in general. This phenomenon is a manifestation of the so-called non-Hermitian skin effect, where eigenstates in the bulk are localized at either end of an open chain. Namely, we should take into account the non-Hermitian skin effect to study the general properties of nonHermitian systems. In Chap. 3, we will discuss in detail the difference between an

22

2 Hermitian Systems and Non-Hermitian Systems

2

|E|

Fig. 2.6 Energy levels in the non-Hermitian Su-Schrieffer-Heeger model with a finite system size under an open boundary condition (black) and under a periodic boundary condition (blue). The green dashed lines represent the parameter t1 where the bulk gap of the periodic chain closes. The red line represents the topological edge states in the open chain. The values of the other parameters are set to be t2 = 1 and γ = 4/3

1

0

-1

0

1

t1 open chain and a periodic chain in a simple non-Hermitian model and will study the non-Hermitian skin effect. Finally, we comment on the results obtained in Ref. [91]. In the non-Hermitian SSH model (2.44), the previous work showed the calculation of energy bands and Brillouin zone under an open boundary condition. Then, the bulk-edge correspondence can be established, which means that the previous work can predict the value of the parameter t1 at which the edge states appear. Nevertheless, the method in this previous work can be applied only to the model considered. Hence, we need to generalize this method in order that one can apply the theory to any non-Hermitian system. We will mention in detail this point in Chap. 4. Appendix The Berry phase is a geometric phase which naturally appears in quantum mechanics when a system is deformed adiabatically. In this Appendix, we describe the basic properties of the Berry phase. First of all, we focus on the Hamiltonian dependent on certain parameters R. Then, the Hamiltonian can be written as H = H (R) .

(2.48)

From the time-independent Schrödinger equation H (R) |ψn (R) = E n (R) |ψn (R),

(2.49)

where E n (R) and |ψn (R) are the energy eigenvalues and the eigenstates of the Hamiltonian (2.48), respectively, and n represents a band index. In the following, we study the adiabatic time evolution of the system. The time-dependent Schrödinger equation satisfies

2.3 Violation of the Bulk-Edge Correspondence in a Non-Hermitian System

i

∂ | (R (t)) = H (R (t)) | (R (t)). ∂t

23

(2.50)

When we assume that the initial state of the wave function | (R (t)) corresponds to the nth eigenstate, i.e., | (R (t = 0)) = |ψn (R (t = 0)), because of the adiabatic time evolution, | (R (t)) can be expressed as | (R (t)) = eiθn (t) |ψn (R (t)).

(2.51)

Hence, from Eqs. (2.50) and (2.50), we can get   dθn (t) ∂ |ψn (R (t)) = E n (R (t)) eiθn (t) |ψn (R (t)), + i eiθn (t) i dt ∂t

(2.52)

and by acting ψn (R (t))| from the left to Eq. (2.52), we can obtain

 ∂ 1 dθn (t) = i ψn (R (t)) ψn (R (t)) − E n (R (t)) , dt ∂t 

(2.53)

and by integrating Eq. (2.53), 1 θn (t) = − 



t

   dt E n R t + i

0



t 0



   ∂    dt ψn R t ψn R t . (2.54) ∂t

Equation (2.54) means that | (R (t)) can be rewritten as    i t    dt E n R t |ψn (R (t)), | (R (t)) = eiγn (t) exp −  0 

where

t

γn (t) = i 0



   ∂    dt ψn R t ψn R t . ∂t

(2.55)



(2.56)

The factor γn (t), called Berry phase, is geometric phase. On the other hand, the other factor  i t    (2.57) dt E n R t −  0 is the classical contribution to the phase. We note that while γn (t) is not gauge invariant in general, it can have gauge invariant in special cases. For convenience, we introduce a quantity called Berry connection; i.e.,

 ∂ ψ , An (R) = i ψn (R) | (R) n ∂R and then, Eq. (2.56) can be rewritten as

(2.58)

24

2 Hermitian Systems and Non-Hermitian Systems

 γn (t) =

t

dt

     t · An R t . ∂t

∂R

0

(2.59)

Here, if we take a closed path C in the parameter space, we finally have  γn =

d R · An (R) .

(2.60)

C

Therefore, the Berry phase γn for a closed path C is now gauge invariant, and it is independent of the time dependence of the parameters. Now, from Eq. (2.58), we also introduce a quantity called Berry curvature as B n (R) = ∇ R × An (R)

 ∂ψn (R) ∂ψn (R) × . =i ∂R ∂R

(2.61)

Then, the Berry curvature satisfies

∂ψn (R) ∂ψn (R) Bn,i (R) = i i jk ∂ R j ∂ Rk



  ∂ψn (R) ∂ψn (R) ψm (R) ψm (R) , = i i jk ∂R ∂R 

m

j

(2.62)

k

where i jk is the Levi-Civita symbol. The contribution from m = n is zero because ψn (R) |ψn (R) = 1. On the other hand, when m = n, Eq. (2.49) implies



∂ψn (R) ∂ψn (R) ∂ E n (R) ∂H (R) = . |ψn (R) + H |ψn (R) + E n (R) ∂ Rk ∂ Rk ∂ Rk ∂ Rk (2.63) Here, by acting ψm (R)| from the left to Eq. (2.63), we can get 





 ∂ψn (R) ∂H (R) 1 ψn (R) , ψn (R) = ψm (R) ∂ Rk En − Em ∂ Rk

(2.64)

and similarly, we have 





 ∂H (R) ∂ψn (R) 1 ψm (R) = ψm (R) . ψn (R) ∂ Rj En − Em ∂ Rj

(2.65)

Finally, from Eqs. (2.62), (2.64), and (2.65), the Berry curvature can be expressed as



∂H (R) ∂H (R) ψ ψ ψ ψ (R) (R) (R) (R) m ∂R m ∂R n  n j k Bn,i (R) = i i jk . 2 − E (E ) n m m=n (2.66) 

2.3 Violation of the Bulk-Edge Correspondence in a Non-Hermitian System

25

In particular, by using the Bloch eigenstates, the momentum representation of the Berry connection and the Berry curvature can be written as 

An (k) = iu n (k) |∇ k |u n (k), B n (k) = ∇ k × An (k) ,

(2.67)

respectively.

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.

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27

Chapter 3

Non-Hermitian Open Chain and Periodic Chain

3.1 Generalized Brillouin Zone and Energy Spectrum in Open Chain First of all, in this section, we study the energy spectrum in a simple non-Hermitian tight-binding model with an open boundary condition. This model is known as the Hatano-Nelson model without disorder [1]. The real-space Hamiltonian of this system is given by L−1    † t R cn+1 (3.1) cn + t L cn† cn+1 , H= n=1

where t R , t L ∈ R are the nearest-neighbor asymmetric hopping amplitudes to the right and to the left, respectively. We show a schematic figure of this system in a finite open chain with the system size L in Fig. 3.1a. The real-space eigen-equation H |ψ = E|ψ, for the eigenvector |ψ = (ψ1 , . . . , ψ L )T , can be written as t R ψn−1 + t L ψn+1 = Eψn , (n = 1, . . . , L), ψ0 = ψ L+1 = 0.

(3.2)

From the theory of linear difference equations, a general solution of the recursion Eq. (3.2) is written as ψn = (β1 )n φ (1) + (β2 )n φ (2) , (3.3) where β j ( j = 1, 2) are the solutions of the characteristic equation t R β −1 + t L β = E.

(3.4)

Together with the open boundary conditions ψ0 = 0 and ψ L+1 = 0, we can get 

1 1 (β1 ) L+1 (β2 ) L+1



φ (1) φ (2)



  0 = 0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_3

(3.5)

29

30

3 Non-Hermitian Open Chain and Periodic Chain

Fig. 3.1 a Schematic figure of the simple model with the system size L. b Schematic figure of the simple model. c-1 Schematic figures of the distribution of β (m) √ and that of E (m) with m = 1, · · · , L. In the limit of L → ∞, β (m)√forms a√ circle with the radius t R /t L , and E (m) forms the energy spectrum in the range of −2 t R t L , 2 t R t L , shown in (c-2)

and obtain a boundary equation, which represents the boundary conditions at the two ends, as   L+1 β1 =1 (3.6) β2 so that the coefficients φ (1) , φ (2) take nonzero values. Then one can get β1 = e2iθm , β2

  mπ , m = 1, . . . , L . θm = L +1

(3.7)

From Eq. (3.4), β j ( j = 1, 2) can be written as β1(m) = r eiθm , β2(m) = r e−iθm ,

(3.8)

√ where r = β1,2 = t R /t L because β1 β2 = t R /t L obtained from the Vieta’s formula. Therefore the eigenstate (3.3) is written as n

n

ψn(m) ∝ r eiθm − r e−iθm ∝ r n sin nθm ,

(3.9)

and, the eigenenergy (3.4) is given by √ E (m) = 2 t R t L cos θm .

(3.10)

From Eqs. (3.9) and (3.10), the distribution of the discrete eigenstates and that of the discrete energy levels are shown in Fig. 3.1c-1. Now, as the system size L becomes larger, these eigenstates and energy levels become dense. Finally, in the limit of L → ∞ [Fig. 3.1b], the energy levels form the energy spectrum as shown in Fig. 3.1c-2. In fact, since Eq. (3.8) leads to the form of β j ( j = 1, 2) as t R iθ t R −iθ β1 = e , β2 = e , (3.11) tL tL

3.1 Generalized Brillouin Zone and Energy Spectrum in Open Chain

31

where θ is a real number, and Eq. (3.4) gives the form of the energy spectrum as √ E = 2 t R t L cos θ.

(3.12)

In comparison with the Hermitian case, which is realized when t R = t L , we can intuitively understand these results from the viewpoint of the conventional Bloch band theory. Equation (3.3) means that β(= β1 , β2 ) can be related with the Bloch wave number k by β = eik . In this sense, the distribution of β(= eik ), which is called generalized Brillouin zone (GBZ), gives a non-Hermitian extension of the Brillouin zone. In a one-dimensional (1D) Hermitian system, the wave number is real, and the GBZ is always a unit circle. On the other hand, in the present case, the GBZ is not a unit circle, meaning that the corresponding wave number is not real in general. We note that the eigenstate is a superposition of two “plane waves” with β1 and β2 , and these two values satisfy Eq. (3.4) with the same energy E. Therefore, once the GBZ is shown, the energy eigenvalues E are calculated As a result, the   √ from√Eq. (3.4). energy spectrum is formed in the range of −2 t R t L , 2 t R t L [Fig. 3.1c-2]. Finally, we comment on the dependence of the above results on boundary conditions. For example, we put the on-site potential a and b at the site n = 1 and at the site n = L, respectively. Then, the form of the boundary Eq. (3.6) in a finite open chain is modified as 

β1 β2

 L+1 =

(aβ1 − t R ) (b/β2 − t L ) . (aβ2 − t R ) (b/β1 − t L )

(3.13)

The energy levels (3.8) are also modified because they are obtained by solving Eqs. (3.4) and (3.13). This means that the energy levels in a finite open chain depend on boundary conditions. As a result, unlike Fig. 3.1c-1, these energy levels deviate from the energy spectrum when the system size is small [Fig 3.2a]. Nevertheless, as the system size becomes larger, these energy levels approach the energy spectrum as shown in Fig. 3.2b. In fact, the energy levels in the limit of a large system size asymptotically match the energy spectrum. This reflects that the GBZ is independent of boundary conditions in an open chain. In general, under any boundary conditions in an open chain, the energy spectrum can be obtained as shown in Fig. 3.1c-2.

Fig. 3.2 Distribution of the energy levels (red dots) in a finite open chain with the on-site potential a and b at the boundary sites. We set the parameter as t R = 3, t L = 1, a = 1.2 + 1.5i, and b = 0.7 + 0.8i, and the system size L as a L = 5 and b L = 60. The black lines represent the energy spectrum

32

3 Non-Hermitian Open Chain and Periodic Chain

3.2 Energy Spectrum in Periodic Chain Next, in this section, we investigate the energy spectrum in the simple model (3.1) with a periodic boundary condition. Firstly, with the system size L, we can calculate the analytical expression of the energy levels by imposing the periodic boundary condition ψ L+1 = ψ1 to the wave function (3.3). Since β can be obtained as β (m) = eiθm ,

  2π m , m = 0, . . . , L − 1 θm = L

(3.14)

from Eq. (3.4), the energy levels can be written as E (m) = t R e−iθm + t L eiθm .

(3.15)

Similar to Sect. 3.1, in the limit of L → ∞, we can get the energy spectrum in a periodic chain as (3.16) E = t R e−iθ + t L eiθ , where θ ∈ R. In this case, β can be written as β = eiθ , θ ∈ R, and we can get the conventional Brillouin zone, similar to Hermitian cases.

3.3 Comparison Between Open Chain and Periodic Chain From Eqs. (3.12) and (3.16), one can see the difference between the energy spectrum under an open boundary condition and that under a periodic boundary condition (Fig. 3.3). Furthermore, we can confirm that the eigenstate (3.9) is exponentially localized at either end of an open chain. On the other hand, in a periodic chain, Eq. (3.14) shows that the eigenstates extend over the whole system. Thus, in nonHermitian systems, the energy eigenvalues and the behavior of the eigenstates dras-

Fig. 3.3 Energy spectrum in an open chain (blue) and in a periodic chain (red)

Im Re

3.3 Comparison Between Open Chain and Periodic Chain

33

tically change between under an open boundary condition and under a periodic boundary condition. The phenomenon that bulk eigenstates are localized at either end of an open chain is called the non-Hermitian skin effect. While the energy spectrum in a periodic chain can be obtained by the straightforward procedure as explained in Sect. 3.2, the calculation of the energy spectrum in an open chain is cumbersome as shown in Sect. 3.1. In Chap. 4, we will extend the procedure to calculate the GBZ and the energy spectrum to general non-Hermitian systems.

3.4 Summary In this chapter, we studied the difference between the energy spectrum under an open boundary condition and that under a periodic boundary condition. We showed that the energy spectrum and the behavior of the eigenstates drastically change between an open chain and a periodic chain because of the non-Hermitian skin effect. In an open chain, we showed a way to calculate the GBZ and the energy spectrum. From the analytical expression, one can see the asymptotic behavior of this system in the limit of a large system size. Furthermore, we emphasized that boundary conditions do not affect the asymptotic behavior of the system with a large system size. This reflects that the GBZ isindependent of any boundary conditions in an open chain.

Reference 1. Hatano N, Nelson DR (1996) Phys Rev Lett 77:570

Chapter 4

Non-Bloch Band Theory of Non-Hermitian Systems and Bulk-Edge Correspondence

4.1 Non-Bloch Band Theory In this section, we explain the non-Bloch band theory in a one-dimensional (1D) nonHermitian tight-binding system. First of all, we explain the concept of this theory and give the main result, i.e., the condition for the generalized Brillouin zone (GBZ). After that, we derive this condition from the equation obtained by an open boundary condition which the eigenstates of a real-space Hamiltonian satisfy.

4.1.1 Concept We can generalize the results obtained in Chap. 3 to any 1D non-Hermitian systems. We start with a 1D tight-binding system with spatial periodicity as shown in Fig. 4.1a. A unit cell is composed of q degrees of freedom, such as sublattices, spins, or orbitals, and the electrons hop to the N th nearest unit cells. Then, its Hamiltonian can be written as q N    † ti,μν cn+i,μ cn,ν , (4.1) H= n

i=−N μ,ν=1

† where cn,μ (cn,μ ) is a creation (an annihilation) operator of an electron with an index μ (μ = 1, . . . , q) in the nth unit cell, and ti,μν is a hopping amplitude to the ith nearest unit cell. This Hamiltonian becomes non-Hermitian when ti,μν is not equal to ∗ . The real-space eigen-equation is written as H |ψ = E|ψ, where the eigent−i,νμ vector is given by |ψ = (. . . , ψ1,1 , . . . , ψ1,q , ψ2,1 , . . . , ψ2,q , . . . )T . As is similar to the model in the previous chapter, |ψ can be represented as a linear combination:

ψn,μ =



 n ( j) ( j) φn,μ , φn,μ = β j φμ( j) , (μ = 1, . . . , q) ,

(4.2)

j

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_4

35

36

4 Non-Bloch Band Theory of Non-Hermitian Systems …

Fig. 4.1 a One-dimensional tight-binding system. A unit cell includes q degrees of freedom, and the range of hopping is N . b Schematic figures of b-1 energy levels in a finite open chain with various system sizes L and of b-2 energy spectra in an open chain. The vertical axis represents the distribution of the complex energy E

where β = β j are the solutions of the characteristic equation defined as det [H (β) − E] = 0, [H (β)]μν =

N 

ti,μν (β)i , (μ, ν = 1, . . . , q) ,

(4.3)

i=−N ( j)

and the coefficients φμ = φμ are given by the eigenvalue equation of the non-Bloch matrix H (β), explicitly rewritten as q 

[H (β)]μν φν = Eφν , (μ = 1, . . . , q) .

(4.4)

ν=1

We note that H (β) becomes the Bloch Hamiltonian if we rewrite it in terms of the conventional Bloch wave number k. Furthermore, Eq. (4.3) is an algebraic equation for β with an even degree 2M = 2q N . Now, we explain the concept of the non-Bloch band theory. As shown in Fig. 4.1b-1, the resulting energy levels are discrete in a finite open chain with the system size L. Here, as L becomes larger, the energy levels become dense and asymptotically continuous. Finally, in the limit of L → ∞, the energy spectra are formed as shown in Fig. 4.1b-2. Here, the asymptotic distribution of β for L → ∞ is the GBZ. We note that in this case, the absolute value of β is not necessarily unity, and the GBZ is obtained as a loop on the complex plane. The key question is how to construct the GBZ of the system considered here. From the argument so far, we may need to calculate the energy levels for finite L and to study its asymptotic behavior for L → ∞. This is a cumbersome procedure, and the result may possibly depend on boundary conditions. Nevertheless, we find a method to calculate the GBZ, without going through calculation on a finite open chain. This largely simplifies the calculation. It is worth noting that the GBZ is independent of boundary conditions. Thus, while energy levels in a finite open chain depend on boundary conditions, their asymptotic behaviors do not. Below, we explain the way to calculate the GBZ, which reproduces the energy spectra. Let β j ( j = 1, . . . , 2M) be the solutions of Eq. (4.3). When we number the 2M solutions so as to satisfy

4.1 Non-Bloch Band Theory

37

|β1 | ≤ |β2 | ≤ · · · ≤ |β2M−1 | ≤ |β2M | ,

(4.5)

we find that the condition for the GBZ is given by |β M | = |β M+1 | .

(4.6)

Namely, the trajectories of β M and β M+1 satisfying Eq. (4.6) give the GBZ. The example in Chap. 3 is a special case with M = 1. In Sect. 4.2, we will show some examples of the GBZ and the energy spectra calculated by the GBZ. Here, we note that although the eigenenergies for the energy spectra are obtained from Eq. (4.3) by putting β = β M and β = β M+1 , the eigenvectors of Eq. (4.4) do not directly express eigenstates of the Hamiltonian (4.1). Instead, the eigenstates of the Hamiltonian (4.1) is given by Eq. (4.2), which involves the terms with β = β1 , . . . , β2M . Nonetheless, the non-Bloch band theory explained here says that the eigenenergies for the energy spectra are determined by β M and β M+1 . Thus, in the calculation of the GBZ, we do not need to solve the eigenvalue problem in Eq. (4.4). The non-Bloch matrix H (β) is introduced here in order to express Eq. (4.3). The condition for the GBZ (4.6) can be regarded as a condition for the formation of a standing wave. Namely, Eq. (4.6) means that the decay lengths of the eigenstates corresponding to β M and β M+1 are equal so that the wave function vanishes at both ends of an open chain. For example, in the simple model (3.1) given in Chap. 3, the wave function (3.9) represents a standing wave apart from the factor r n , as a superposition of two counterpropagating “plane waves”. Furthermore, Eq. (4.6) is physically reasonable in several aspects. Firstly, this condition does not depend on any boundary conditions in an open chain. Secondly, in the Hermitian limit, we can rewrite Eq. (4.6) to the well-known result, i.e., |β M | = |β M+1 | = 1. For example, in the simple model (3.1) with the case of t R = t L , the system becomes Hermitian, and the GBZ becomes a unit circle, identified with the conventional Brillouin zone. The rigorous proof is given in Appendix in this chapter. Finally, we mention the cases that Eq. (4.3) is a reducible algebraic equation. In such cases, it can be factorized as det [H (β) − E] = f 1 (β, E) · · · f m (β, E), where f i (β, E) (i = 1, . . . , m) are algebraic equations for β and E. For simplicity, we assume that they are algebraic equations for β with 2Mi . In this  degree  an  even  case, the GBZ can be obtained from the conditions β Mi  = β Mi +1  (i = 1, . . . , m), instead of Eq. (4.6).

4.1.2 Condition for the Generalized Brillouin Zone In this subsection, we derive the condition (4.6). To this end, we focus on boundary conditions in a finite open chain with L unit cells. The wave functions are written as ψn,μ =

2M   n ( j) β j φμ , (n = 1, . . . L , μ = 1, . . . , q). j=1

(4.7)

38

4 Non-Bloch Band Theory of Non-Hermitian Systems …

The equations obtained from boundary conditions include the 2q M unknown vari( j) ables φμ . The real-space eigen-equation H |ψ = E|ψ fixes the ratio between the ( j) values of φμ sharing the same value of j. Therefore, we can reduce the 2q M vari( j) ables to the 2M variables φμ with a single value of μ, e.g., μ = 1. As a result, we ( j) can get M equations for the 2M variables φ1 ( j = 1, . . . , 2M) at the left end of an open chain around n = 1 as 2M 

  ( j) f i β j , E, S φ1 = 0, (i = 1, . . . , M) ,

(4.8)

j=1

and we can also get M equations at the right end of an open chain around n = L as 2M 

    L ( j) gi β j , E, S β j φ1 = 0, (i = 1, . . . , M) ,

(4.9)

j=1

where S is a set of the system parameters ti,μν , and f i and gi are functions of β j , E, and S. We note that both f i and gi depend on boundary conditions but not on the system size L. We emphasize here that any kinds of boundary conditions for an open chain can be written as M equations for the left end and M equations for the right end. By combining Eqs. (4.8) and (4.9), we can obtain a condition for the existence of nontrivial solutions for φ1(1) , . . . , φ1(2M) as   f 1 (β1 , E, S)   ..   .   f M (β1 , E, S)    g (β , E, S) (β ) L  1 1 1   .  ..    g (β , E, S) (β ) L M

1

1

··· .. . ··· ··· .. . ···

    ..   .   f M (β2M , E, S)   = 0. L  g1 (β2M , E, S) (β2M )    ..  .   L g (β , E, S) (β ) f 1 (β2M , E, S)

M

2M

(4.10)

2M

Then, Eq. (4.10) can be expressed as  P,Q

  F βi∈P , β j∈Q , E, S (βk ) L = 0.

(4.11)

k∈P

The sets P and Q are two disjoint subsets of the set {1, . . . , 2M} such that the number of elements of each subset is M. In this way, the sum included in Eq. (4.11) is taken over all the sets P and Q. By solving Eq. (4.11), we can calculate eigenenergies in an open chain. However, this equation for a general system size cannot be analytically solved. Nonetheless, our aim is to see how the solutions for a large system size form the energy spectra. Therefore, we suppose the system size L to be quite large and investigate a condition to form densely distributed energy levels.

4.1 Non-Bloch Band Theory

39

We now study asymptotic behavior of the solutions of Eq. (4.11) for a large system size L. When |β M | = |β M+1 |, there is only one leading term proportional to (β M+1 · · · β2M ) L in the limit of L → ∞. Then it leads to   F βi∈P , β j∈Q , E, S = 0

(4.12)

with P = {M + 1, . . . , 2M} and Q = {1, . . . , M}. This equation does not depend on L and does not allow the energy spectra. On the other hand, when |β M | = |β M+1 |, there are two leading terms proportional to (β M β M+2 · · · β2M ) L and to (β M+1 β M+2 · · · β2M ) L . Hence, Eq. (4.11) can be written as 

βM β M+1

L

  F βi∈P0 , β j∈Q 0 , E, S  =−  F βi∈P1 , β j∈Q 1 , E, S

(4.13)

with P0 = {M + 1, . . . , 2M} , Q 0 = {1, . . . , M} , P1 = {M, M + 2, . . . , 2M}, and Q 1 = {1, . . . , M − 1, M + 1}. In such a case, the relative phase between β M and β M+1 can be changed almost continuously for a large L, producing the energy spectra. In conclusion, we get the condition for the GBZ as Eq. (4.6). We note  that  while Refs. [1, 2] proposed that the form of the energy spectra requires |βi | = β j  for ∀i, j, this is not sufficient to guarantee the energy spectra. We find that |β M | = |β M+1 | is a sufficient condition. In Sect. 4.2.1, we derive Eq. (4.6) for the non-Hermitian SSH model as an example. In this model, Eq. (4.11) coming from a boundary condition is given by Eq. (4.21).

4.2 Model In this section, we apply the non-Bloch band theory to some models. We reproduce the derivation of the condition for the GBZ in the non-Hermitian SSH model and show that this condition is independent of any boundary conditions in an open chain. Furthermore, we reveal unique non-Hermitian properties.

4.2.1 Non-Hermitian Su-Schrieffer-Heeger Model In this subsection, in terms of the non-Bloch band theory, we investigate the non-Hermitian Su-Schrieffer-Heeger (SSH) model proposed in Refs. [3, 4]. The schematic figure of the non-Hermitian SSH model is shown in Fig. 4.2a, and the real-space Hamiltonian is written as H =



 † γ1  † γ2  † cn,A cn,B + t2 − c t3 cn,A cn+1,B + t1 + cn,B 2 2 n+1,A n



γ2  † γ1  † † cn,B cn+1,A + t1 − cn,B cn,A + t3 cn+1,B + t2 + cn,A . (4.14) 2 2

40

4 Non-Bloch Band Theory of Non-Hermitian Systems …

Fig. 4.2 a Schematic figure of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. The dotted boxes indicate the unit cell. b Non-Hermitian SSH model with the compex potentials a and b on the boundary sites. c-e Generalized Brillouin zone in the non-Hermitian SSH model. The values of the parameters are c t1 = 1.1, t2 = 1, t3 = 1/5, γ1 = 4/3, and γ2 = 0; d t1 = 0.3, t2 = 1.1, t3 = 1/5, γ1 = 0, and γ2 = −4/3; and e t1 = −0.3, t2 = 0.5, t3 = 1/5, γ1 = 5/3, and γ2 = 1/3

Henceforth, we set all the parameters to be real. We note that the system  becomes Hermitian when γ1 = γ2 = 0. For the real-space eigenvector |ψ = . . . , ψ1,A , ψ1,B , ψ2,A , ψ2,B , . . . , we can explicitly write the real-space eigen-equation as ⎧



γ1  γ2  ⎪ ψn,B + t2 − ψn−1,B = Eψn,A , ⎨ t3 ψn+1,B + t1 + 2 2  



γ γ ⎪ ⎩ t2 + 2 ψn+1,A + t1 − 1 ψn,A + t3 ψn−1,A = Eψn,B . 2 2

(4.15)

Here, we can take a general ansatz for the wave function as a linear combination as 

ψn,A ψn,B



  n = βj j



( j)

φA ( j) φB

 ,

(4.16)

where β = β j are the solutions of the characteristic equation det [H (β) − E] = 0 for the non-Bloch matrix

γ1 

γ2  −1 ⎞ + t2 − β t3 β + t1 + ⎜ ⎟ 2 2 H (β) = ⎝



 ⎠, γ2 γ1 −1 β + t1 − + t3 β t2 + 0 2 2 ⎛

0

(4.17) ( j)

and the coefficients φμ = φμ (μ = A, B) are given by the eigenvalue equation of H (β) as 

φA H (β) φB In this case, the characteristic equation



 =E

φA φB

 .

(4.18)

4.2 Model

41



γ1 

γ2  −1

γ2  γ1  + t2 − β β + t1 − + t3 β −1 − E 2 = 0 t3 β + t1 + t2 + 2 2 2 2

(4.19) is a quartic equation for β, having four solutions βi (i = 1, . . . , 4) satisfying |β1 | ≤ |β2 | ≤ |β3 | ≤ |β4 |. In the case of M = 2, the condition for the GBZ is given by |β2 | = |β3 | .

(4.20)

It is obtained by imposing that Eq. (4.16) satisfying an open boundary condition forms a dense set of solutions at L → ∞. As emphasized earlier, Eq. (4.20) does not depend on boundary conditions in an open chain, and we show this below.

4.2.1.1

Derivation of the Condition for the Generalized Brillouin Zone

In order to derive Eq. (4.20), we study the non-Hermitian SSH model with L unit cells with the complex potential a and b on the boundary sites as shown in Fig. 4.2b. Then, we show that the condition for the GBZ is independent of a, b, and L. We write down the boundary conditions for ψn,μ as ⎧

γ1  ⎪ t ψ1,B + t3 ψ2,B + aψ1,A = Eψ1,A , + ⎪ 1 ⎪ ⎪ 2 ⎪ ⎪ 



⎪ γ1 γ2  ⎪ ⎪ ψ1,A + t2 + ψ2,A = Eψ1,B , ⎨ t1 − 2 2 



γ2 γ1  ⎪ ⎪ ψ ψ L ,B = Eψ L ,A , t − + t + ⎪ 2 L−1,B 1 ⎪ ⎪ 2 2 ⎪ 

⎪ ⎪ γ ⎪ ⎩ t3 ψ L−1,A + t1 − 1 ψ L ,A + bψ L ,B = Eψ L ,B . 2

(4.21)

By substituting the general solution ψn,μ =

4   n ( j) β j φμ , (μ = A, B)

(4.22)

j=1 ( j)

to Eq. (4.21), we can obtain four equations for the eight coefficients φμ ( j = 1, . . . , 4, μ = A, B). By recalling that these coefficients satisfy E

( j)

φB =

(t2 −

γ2 /2) β −1 j

+ (t1 + γ1 /2) + t3 β j

( j)

φA

(4.23)

from the eigenvalue Eq. (4.18), we can reduce the problem into four linear equa( j) tions for the four coefficients φA ( j = 1, . . . , 4). Then, the condition for the linear equation to have nontrivial solutions is written as

42

4 Non-Bloch Band Theory of Non-Hermitian Systems …

   1 1 1 1      f2 f3 f4   f1    X β L X β L X β L X β L  = 0,  1 1 2 2 3 3 4 4     g βL g βL g βL g βL  1 1 2 2 3 3 4 4

(4.24)

where X j , f j and g j ( j = 1, . . . , 4) are defined as Xj =

βj (t2 −

γ2 /2) β −1 j

+ (t1 + γ1 /2) + t3 β j

,

f j = aβ j − (t2 − γ2 /2) Eβ −1 j X j, g j = bEβ −1 j X j − (t2 + γ2 /2) β j ,

(4.25)

respectively. Furthermore, Eq. (4.24) can be explicitly written as   L  1 sgn (σ ) f σ (1) − f σ (2) gσ (3) X σ (4) βσ (3) βσ (4) = 0, 2 σ

(4.26)

where the sum is taken over all the permutations σ for four objects. Next, we demonstrate that the condition for the GBZ is given by |β2 | = |β3 |. We have to impose this condition to get the energy spectra when the solutions of Eq. (4.26) are densely distributed for a large system size. When |β2 | = |β3 |, the only leading term in Eq. (4.26) is the term proportional to (β3 β4 ) L , leading to the equation ( f 1 − f 2 ) (g3 X 4 − g4 X 3 ) = 0

(4.27)

in the limit of L → ∞. By combining this equation with Eq. (4.19), the eigenenergies are restricted to discrete values, and it cannot represent the energy spectra. On the other hand, when |β2 | = |β3 | is satisfied, Eq. (4.26) has two leading terms, (β2 β4 ) L and (β3 β4 ) L for a large system size, and Eq. (4.26) can be rewritten as 

β2 β3

L =

( f 1 − f 2 ) (g4 X 3 − g3 X 4 ) . ( f 1 − f 3 ) (g4 X 2 − g2 X 4 )

(4.28)

In the limit of L → ∞, this equation allows a dense set of solutions when the relative phase between β2 and β3 is continuously changed. Therefore, we conclude that Eq. (4.20) is an appropriate condition for the GBZ. We emphasize that Eq. (4.20) is now independent of any boundary conditions; i.e., the complex potential a and b, and the system size L. If we change the value of the parameters, Eq. (4.26) changes; nonetheless, Eq. (4.20) remains the same, and together with Eq. (4.19), we can obtain the energy spectra. Here, we mention the condition for the GBZ for some special cases in the nonHermitian SSH model. For example, when t2 = −γ2 /2, the characteristic Eq. (4.19) becomes a cubic equation for β. Even in this case, we can still regard this equation as a quartic equation by putting t2 + γ2 /2 to be an infinitesimal. Namely, by adding

4.2 Model

43

another solution β = ∞ to three solutions for this cubic equation, our result holds good with four solutions β1 , β2 , β3 , β4 (= ∞). Thus, in general, when the degree of the characteristic equation is less than 2M, we can treat it as a limiting case of an algebraic equation of 2Mth degree by formally adding a solution β = 0 or β = ∞, depending on systems, and the condition for the GBZ remains valid. Another special case is the case of t3 = 0 in the non-Hermitian SSH model. In this case, Eq. (4.19) becomes a quadratic equation for β, and it has two solutions. Even in this special case, if the infinitesimally small hopping amplitude t3 is added, this quadratic equation has the four solutions β = β1 , β2 , β3 , β4 satisfying Eq. (4.5). Then, the above condition corresponds to the condition β1 (= 0) ≤ |β2 | = |β3 | ≤ β4 (= ∞) in the limit of t3 → 0, meaning that the two solutions (β2 , β3 ) of this equation have the same absolute values. In fact, this result agrees with Ref. [1]. 4.2.1.2

Generalized Brillouin Zone

The trajectories of β2 and β3 give the GBZ as shown in Fig. 4.2 c-e with various values of the parameters. It is worth mentioning some features of the GBZ in the following. Firstly, as shown in Fig. 4.2e, |β| on the GBZ takes both values more than 1 and values less than 1 [5–7]. Here |β| > 1 (|β| < 1) means that the eigenstate is localized at the right (left) end of an open chain, representing the non-Hermitian skin effect. Secondly, the GBZ is a closed loop encircling the origin on the complex plane [6, 7]. Finally, the GBZ can have the cusps, corresponding to the cases where three solutions of Eq. (4.19) share the same absolute value. In order to explain the appearance of the cusps  on  the GBZ in detail, we investigate what happens if we impose a condition |βi | = β j  for some i and j among the four solutions instead. We show the trajectories of β2 and β3 by imposing the condition |β2 | = |β3 | in Fig. 4.3a. We also  show  the trajectories of all the solutions of Eq. (4.19) by imposing condition |βi | = β j  for some i and j in Fig. 4.3b. By the construction,

Fig. 4.3 Trajectories of the solutions of the characteristic   Eq. (4.19). a Trajectories of β2 and β3 with |β2 | = |β3 | and b those of βi and β j with |βi | = β j  (i = j). The former corresponds to the generalized Brillouin zone. The parameters are set to be t1 = 0.3, t2 = 0.5, t3 = 1/5, γ1 = 5/3, and γ2 = 1/3

44

4 Non-Bloch Band Theory of Non-Hermitian Systems …

the curve in Fig. 4.3a is a subset of that in Fig. 4.3b. We note that Fig. 4.3b consists of smooth curves, while Fig. 4.3a does not. As seen in Fig. 4.3a, the GBZ has the cusps. They appear when three of the four solutions of Eq. (4.19) share the same absolute value. Suppose |β1 | < |β2 | = |β3 | < |β4 |, and as we go along the GBZ, in fact, |β1 | approaches |β2 | (= |β3 |). At last, when |β1 | = |β2 | = |β3 |, the behavior of the solutions satisfying |β2 | = |β3 | changes, and the cusp appears on the GBZ, as can be seen from Fig. 4.3a and b.

4.2.1.3

Calculation Methods

We can get the energy spectra to judge the condition (4.20) for a given value of E. To explain this, we set the values of the parameters as (t1 , t2 , t3 , γ1 , γ2 ) = (3/10, 1/2, 1/5, 5/3, 1/3). For example, when we take E = E 1 = 0.140 + 0.755i, we obtain the four solutions of the characteristic Eq. (4.19) as (β1 , β2 , β3 , β4 ) = (0.142 − 0.348i, 0.064 + 0.535i, −0.506 + 0.184i, −4.57 − 0.371i) , (|β1 | , |β2 | , |β3 | , |β4 |) = (0.376, 0.538, 0.538, 4.58) .

(4.29) Since these solutions satisfy the condition for the GBZ, we conclude that E 1 is included in the energy spectra. On the other hand, when we substitute E = E 2 = 0.180 + 1.23i into Eq. (4.19), we obtain the solutions as (β1 , β2 , β3 , β4 ) = (0.036 − 0.228i, −0.052 + 0.233i, −1.52 + 1.85i, −3.33 − 1.85i) , (|β1 | , |β2 | , |β3 | , |β4 |) = (0.230, 0.238, 2.39, 3.81) .

(4.30) In this case, E 2 is not in the energy spectra because |β2 | = |β3 |. In this way, from Eq. (4.20), we calculate the energy spectra as shown in Fig. 4.6c. In fact, we can confirm that this result agrees with the energy levels in a finite open chain as shown in Fig. 4.7d. Additionally, we demonstrate the calculation of the GBZ from Eq. (4.19). We first express Eq. (4.19) as E 2 = f (β). Suppose the two solutions β and β have the same absolute values: |β| = β . Then, we have β = βeiθ ,

(4.31)

where θ is real. By taking the difference between two equations:   E 2 = f (β) , E 2 = f βeiθ , we get

(4.32)

4.2 Model

45





γ2  γ1 

γ2 

γ1  −1

1 − e−iθ t3 β −2 1 − e−2iθ + t1 − t2 − + t1 + t3 β 0 = t2 − 2 2 2 2



 γ1  γ1 

γ2 

γ2  + t1 − t3 + t1 + t2 + β 1 − eiθ + t2 + t3 β 2 1 − e2iθ . 2 2 2 2

(4.33) This equation allows us to compute β for a given  value of θ ∈ (0, 2π ). Then we obtain a set of the values of β that satisfies |β| = β . To obtain the GBZ, we should further constrain the values of β by Eq. (4.20). Namely, the absolute values of β and β should be the second and third largest ones among the four solutions. By selecting the values of β and β satisfying this condition, we can get the GBZ.

4.2.2 Ladder Model I In this subsection, in terms of the non-Bloch band theory, we investigate the 1D nonHermitian tight-binding ladder model proposed in Ref. [8], as shown in Fig. 4.4a. First of all, we give the real-space Hamiltonian in this model as  ir

  † † † † † † cn+1,A v cn,A cn,B + cn,B cn,A + cn,A − cn,A cn+1,A − cn+1,B cn,B + cn,B cn+1,B 2 n  iγ

 r † † † † † † cn+1,A cn,B + cn,B cn,A cn+1,A + cn+1,B cn,A + cn,A cn+1,B + cn,A − cn,B cn,B , + 2 2

H=

(4.34) where we take all the parameters to be real, and henceforth, we set |r/γ | ≤ 1/2. When the eigenvector of the real-space eigen-equation is written as |ψ =  T . . . , ψ1,A , ψ1,B , ψ2,A , ψ2,B , . . . , the equation for ψn,μ can be written as

Fig. 4.4 a Schematic figure of the tight-binding ladder model proposed in Ref. [8]. This model includes a gain on sublattice A and a loss on sublattice B. b Energy levels of the ladder model in a finite open chain. We 1/3 and the system size as L = 50. The gap closes √set the parameter as r/γ = √ at v/γ = ±c1 = ± 13/6 and v/γ = ±c2 = ± 5/6. The topological edge states are show in red

46

4 Non-Bloch Band Theory of Non-Hermitian Systems …

⎧     ⎪ ⎪ ir ψn−1,A − ψn+1,A + iγ ψn,A + r ψn−1,B + ψn+1,B + vψn,B = Eψn,A , ⎨ 2 2 2 (4.35)    ⎪ ir iγ r  ⎪− ⎩ ψn−1,B − ψn+1,B − ψn,B + ψn−1,A + ψn+1,A + vψn,A = Eψn,B . 2 2 2 Here, we can take a general ansatz for the wave function as a linear combination as 

ψn,A ψn,B



  n = βj j



( j)

φA ( j) φB

 ,

(4.36)

where β = β j are the solutions of the characteristic equation det [H (β) − E] = 0 for the non-Bloch matrix     iγ ir  r  β + β −1 σx + − β − β −1 + σz , (4.37) H (β) = v + 2 2 2 where σi (i = x, y, z) are the Pauli matrices. Then, the characteristic equation is explicitly written as  



γ  −1 γ2 γ 2 2 2 β + r +v − − E +r v − β = 0, r v+ 2 4 2

(4.38)

and therefore, the condition for the GBZ is given by |β1 | = |β2 | since Eq. (4.38) is a quadratic equation for β. In this case, the GBZ becomes a circle with the radius   R ≡ β1,2  =

   v − γ /2     v + γ /2 ,

(4.39)

which is given by the Vieta’s formula. We find that this tight-binding model can be regarded as the non-Hermitian SSH model. Indeed, by the unitary transformation σx → σx , σ y → −σz , σz → σ y ,

(4.40)

the non-Bloch matrix (4.37) can be rewritten as  



γ γ H (β) → v + + rβ −1 σ+ + v − + rβ σ− , (4.41) 2 2   where σ± = σx ± iσ y /2. Therefore, the form of this matrix completely corresponds to that of Eq. (4.17) with the case of γ2 = t3 = 0. Thus, in this non-Hermitian ladder model, we can expect the appearance of the topological edge states originating from the Hermitian SSH model. To see this, we calculate the parameters where the two bands touch at E = 0 because the two energy eigenvalues E 1 and E 2 are opposite in sign, E 1 = −E 2 , because of the form of

4.2 Model

47

Eq. (4.38). By substituting β = Reiϕ , ϕ ∈ R to Eq. (4.38), we can get the parameters where the gap closes as   2 r v 1 =± ± . (4.42) γ 4 γ Therefore, the zero-energy topological edge states are expected to appear in   the parameter region v/γ ∈ − 1/4 + (r/γ )2 , − 1/4 − (r/γ )2 and v/γ ∈   1/4 − (r/γ )2 , 1/4 + (r/γ )2 . Indeed, we confirm the appearance of the topological edge states in a finite open chain with r/γ = 1/3 within these region as shown in Fig. 4.4b. Thus, by the non-Bloch band theory, the bulk-edge correspondence in non-Hermitian systems is fully shown in the same sense as in Hermitian systems. In Sect. 4.3, we will discuss in detail the bulk-edge correspondence in the non-Hermitian SSH model by defining a topological invariant from the GBZ.

4.2.3 Ladder Model II The previous work proposed that in a non-Hermitian multi-band system, each of energy bands is associated with its own GBZ [7]. On the other hand, in a Hermitian system, all energy bands have common GBZ, being a unit circle β = eik , k ∈ R. In this subsection, we show the splitting of the GBZ in the two-band non-Hermitian model as shown in Fig. 4.5a. We assume no symmetries except for a translation symmetry. The real-space Hamiltonian of this system can be written as H=

 † † † t R cn+1,A cn,A + cn,A cn,B + t L cn+1,A cn,A n

† +t R cn+1,B cn,B

 † † † + cn,B cn,A + U cn,B cn,B + t L cn,B cn+1,B .

(4.43)

By the same procedure in the previous sections, we can get the non-Bloch matrix as

Fig. 4.5 a Schematic figure of the ladder model II b Generalized Brillouin zone and c energy spectra in the ladder model II. In b and c, the same bands are shown in the same color. We set the values of the parameters as t R = 2, t L = 1, t R = 1, t L = 3, and U =  = −1

48

4 Non-Bloch Band Theory of Non-Hermitian Systems …

 H (β) =

t R β −1 + t L β





t R β −1 + U + t L β

 .

(4.44)

By applying the condition (4.6) to the solutions of the characteristic equation det [H (β) − E] = 0, we can get the GBZ and the energy spectra. The results are shown in Fig. 4.5b and c. One can see that the GBZ splits into two curves, each of which corresponds to the individual band. We note that under some additional symmetries, some bands necessarily share the same GBZ. For example, the non-Hermitian SSH model (4.14) has only one GBZ because it has a sublattice symmetry (SLS). Here, the SLS is defined as H −1 = −H for a real-space Hamiltonian, where is a unitary matrix satisfying 2 = +1. Namely, the eigenenergy of such systems appears in pairs (E, −E), both of which come from the same GBZ.

4.3 Bulk-Edge Correspondence In this section, we establish the bulk-edge correspondence in non-Hermitian systems in terms of the non-Bloch band theory. First of all, in a 1D non-Hermitian system with the SLS, we define a topological invariant, called winding number, from the GBZ. Next, we demonstrate that in the non-Hermitian SSH model, the appearance of the topological edge states corresponds to the region where the winding number takes nonzero values.

4.3.1 Winding Number Let us start with a 1D non-Hermitian tight-binding system with the SLS. In this system, we can define a topological invariant, called winding number, from the GBZ. The formula of the winding number can be derived in the following manner. First of all, by the procedure explained in Sect. 4.1.1, we can get the non-Bloch matrix from the real-space Hamiltonian (4.1) in a large open chain. Then, if we put the matrix form of the SLS as   1l 0 ≡ , (4.45) 0 −1l we can rewrite the non-Bloch matrix of this system as the block off-diagonal form as  H (β) =

 0 R+ (β) , R− (β) 0

(4.46)

4.3 Bulk-Edge Correspondence

49

where β ≡ eik , k ∈ C, 1l is an N × N identity matrix in a 2N band system, and R± (β) are N × N matrices. In the following, we assume that this system has a gap around E = 0, but without exceptional points. Furthermore, since this system always has pairs of the eigenenergies (E, −E) due to the SLS, we can assume that the bands are composed of N occupied bands with E = −E i (i = 1, . . . , N ) and N unoccupied bands with E = E i (i = 1, . . . , N ). The eigenvalue equations of the non-Bloch matrix (4.46) for the right and left eigenvectors  |ψ R (β) =

|a R (β) |b R (β)



  , ψ L (β)| = a L (β)|, a R (β)|

(4.47)

are given by R+ (β) |b R (β) = E|a R (β), R− (β) |a R (β) = E|b R (β),

(4.48)

a L (β)|R+ (β) = E b L (β)|, b L (β)|R− (β) = E a L (β)|,

(4.49)

and

respectively. For the right and left eigenvectors, we can reduce Eqs. (4.48) and (4.49) to (4.50) R+ (β) R− (β) |a R (β) = E 2 |a R (β) and

a L (β)|R+ (β) R− (β) = E 2 a L (β)|,

(4.51)

respectively. Here, we introduce the right and left eigenvectors of the N × N matrix R+ (β) R− (β) as |a R/L ,1 (β), . . . , |a R/L ,N (β), respectively, and the eigenvalues as E 12 (β) , . . . , E N2 (β). Furthermore, the right and left eigenvectors can satisfy    a L ,i (β) a R, j (β) = δi j

(4.52)

by taking the biorthogonal basis [9]. At last, we can obtain the biorthogonal eigenvectors of the non-Bloch matrix (4.46) in the occupied bands for i = 1, . . . , N as ⎛ 1 |ψ R,i (β) = √ ⎝ 2 −

|a R,i (β)

⎞

⎠ 1 R− (β) |a R,i (β) E i (β)

(4.53)

and 1

ψ L ,i (β)| = √ 2



a L ,i (β)|, −

 1

a L ,i |R+ (β) . E i (β)

(4.54)

50

4 Non-Bloch Band Theory of Non-Hermitian Systems …

R/L ,i (β) = σz |ψ R/L ,i (β). We note that the unoccupied eigenvectors are given by |ψ Then, the Q matrix can be defined as Q (β) =

N    R,i (β) ψ L ,i (β)| − |ψ R,i (β) ψ L ,i (β)| , |ψ

(4.55)

i=1

and in the matrix form, it can be written as Q (β) =

N  i=1

1 E i (β)



 O |a R,i (β) a L ,i (β)|R+ (β) . O R− (β) |a R,i (β) a L ,i (β)| (4.56)

Here, we define the matrices q (β) and q −1 (β) as q (β) ≡

N  i=1

and q −1 (β) ≡

N  i=1

1 |a R,i (β) a L ,i (β)|R+ (β) E i (β)

(4.57)

1 R− (β) |a R,i (β) a L ,i (β)|, E i (β)

(4.58)

respectively.

4.3.1.1

Multi-band Model

From the above result, in a simple case, we can get the formula of the winding number on the analogy in Hermitian systems [10]. Namely, if the N occupied bands share the same GBZ denoted as Cβ(all) , it can be written as w=

i 2π

=

i 2π

Cβ(all)

=

i 2π

Cβ(all)

Cβ(all)

! " Tr dq q −1 (β) ! " dTr log q (β) d log det [q (β)] .

(4.59)

In fact, in Hermitian cases, Eq. (4.59) is reduced to the form of Eq. (2.24) in the case of n = 0 because the GBZ becomes a unit circle, which means that the Bloch wave number takes real values. On the other hand, as mentioned in Sect. 4.2.3, each occupied band can have their own GBZ denoted as Cβ,i (i = 1, . . . , N ). In this case,

4.3 Bulk-Edge Correspondence

51

we conjecture the formula of the winding number as [7] N i  w = 2π i=1

=

i 2π

Cβ,i

N  i=1

! " Tr dqi qi−1 (β) d log det [qi (β)] ,

(4.60)

Cβ,i

where qi (β) =

1 |a R,i (β) a L ,i (β)|R+ (β) , (i = 1, . . . , N ). E i (β)

(4.61)

However, this formula is not correct because Eq. (4.60) cannot be reduced to the form of Eq. (4.59) due to # N $ N   det qi (β) = det [qi (β)] . (4.62) i=1

i=1

So far, the correct formula of the winding number in a 1D non-Hermitian system with the SLS with multi-bands has not been revealed. It is left as a future work.

4.3.1.2

Two-Band Model

On the other hand, we can write the winding number into a brief form in a two-band model because the two bands share only one GBZ. For the non-Bloch matrix H (β) = R+ (β) σ+ + R− (β) σ− ,

(4.63)

  where σ± = σx ± iσ y /2, the eigenvalues are given by  E ± (β) = ± R+ (β) R− (β), and the right and left eigenvectors can be written down as   1 √ R+ |u R,+  = √ √ , R+ R− 2 R+ R−   1 √R+ , |u R,−  = √ √ 2 R+ R− − R+ R−   √ 1 R− , R+ R− ,

u L ,+ | = √ √ 2 R+ R−   √ 1 R− , − R+ R− ,

u L ,− | = √ √ 2 R+ R−

(4.64)

(4.65)

52

4 Non-Bloch Band Theory of Non-Hermitian Systems …

respectively. The subscript +/− means that the eigenvectors with +/− have the eigenvalues E + or E − , respectively. From Eq. (4.65), the Q matrix is given by Q (β) = |u R,+ (β) u L ,+ (β)| − |u R,− (β) u L ,− (β)|   1 0 R+ (β) . = √ R+ (β) R− (β) R− (β) 0 √ Therefore, we obtain q = R+ / R+ R− , and the winding number as i 2π i = 2π

dq q −1 (β)

w=

=− =−

Cβ

d log q (β) Cβ

" 1 ! arg q (β) Cβ 2π ! " 1 arg R+ (β) − arg R− (β) Cβ 2π

(4.66)

2

.

(4.67)

Thus, the winding number is determined by the change in the phase of the functions R± (β) when β goes along the GBZ denoted as Cβ in a counterclockwise way. Obviously, when the system becomes Hermitian, Eq. (4.67) is reduced to the form of Eq. (2.31). Importantly, we can show the bulk-edge correspondence between the winding number and the existence of topological edge states as demonstrated in Sect. 4.3.2. Here, in order to investigate how to determine the values of the winding number, let ± denote the loops drawn by R± (β) when β goes along the GBZ in a counterclockwise way. Then, the value of the winding number is determined by the number of times that ± surrounds the origin on the complex plane. When neither + nor − surrounds the origin, the winding number is zero. On the other hand, the winding number takes a nonzero value when both + and − simultaneously surround the origin. We comment on a special case in the two-band model. Suppose we change the system parameters continuously. Then, at some values of the parameters, only one of + and − can pass the origin on the complex plane. In this case, since only one of R+ (β) or R− (β) becomes 0, the non-Bloch matrix (4.63) is written as the Jordan normal form. This indicates that the system has exceptional points [11]. We note that the winding number (4.67) is not well defined because the gap closes. We will discuss in detail this point in Chap. 5.

4.3.2 Phase Diagram in the Non-Hermitian Su-Schrieffer-Heeger Model Now we demonstrate the bulk-edge correspondence for the winding number w defined in Eq. (4.67) in the non-Hermitian SSH model. With the values of the param-

4.3 Bulk-Edge Correspondence

53

Fig. 4.6 Bulk-edge correspondence in the non-Hermitian Su-Schrieffer-Heeger model with the values of the parameters as t3 = 1/5, γ1 = 5/3, and γ2 = 1/3. a Phase diagram on the t1 -t2 plane. The blue and the white regions represent the topological insulator (TI) phase with the winding number w being 1 and the normal insulator (NI) phase with w = 0, respectively. The orange region represents a gapless phase. b Trajectories + (red) and − (blue) on the R plane with t1 = 1 and t2 = 1.4. The arrows mean the direction of the change of the functions R± (β) as β goes in a counterclockwise manner along the generalized Brillouin zone (GBZ). c Energy spectra calculated from the GBZ. We show the results for them along the black arrow in a with t2 = 1.4. d Energy levels in an open chain with the system size L = 100. The red line represents the topological edge states

eters as (t3 , γ1 , γ2 ) = (1/5, 5/3, 1/3), we obtain the phase diagram on the t1 –t2 plane as shown in Fig. 4.6a. In this phase diagram, the white region represents the normal insulator (NI) with w = 0, and the blue region does the topological insulator (TI) phase with w = 1. For example, at the red dot in Fig. 4.6a, from the trajectories ± as shown in Fig. 4.6b, one can find that the value of the winding number is 1 since both + and − surround simultaneously the origin on the R plane. Hence, we expect that the topological edge states appear in the TI phase. In fact, in the energy levels in a finite open chain, we can confirm the appearance of the zero-energy edge states (red in Fig. 4.6d) as expected. We note that the energy spectra calculated from the GBZ [Fig. 4.6c] agree with these energy levels except for the topological edge states. In conclusion, we can establish the bulk-edge correspondence between the topological invariant defined by the GBZ and the existence of the topological edge states in the non-Hermitian SSH model. From another perspective, we further discuss the bulk-edge correspondence in the non-Hermitian SSH model with the values of the parameters as (t1 , t2 , t3 ) = (0, 1, 1/5). In this case, we also obtain the phase diagram in this model as shown in Fig. 4.7a. One can see from this phase diagram that we can continuously change the values of the parameters to the Hermitian limit, γ1 , γ2 → 0, while keeping the gap open. We note that in the case with γ1 = γ2 = 0, the system is topologically nontrivial, and it has the zero-energy topological edge states. By following the proof in Ref. [12], it is shown that the system within the blue region including the origin on the γ1 -γ2 plane has the zero-energy edge states. This is because when the system parameters can continuously change without closing the bulk gap, the winding number does not change, and the topology of the system remains invariant. Therefore, we can prove the bulk-edge correspondence for the non-Hermitian SSH model, and the existence of zero-energy edge states is derived. In fact, we confirm that as shown in Fig. 4.7b, the topological edge states appear in a finite open chain in the regions where the winding number takes 1 (e.g., see Fig. 4.7c). We emphasize that these topological edge states originate from the Hermitian topology. On the other hand,

54

4 Non-Bloch Band Theory of Non-Hermitian Systems …

Fig. 4.7 Phase diagram and bulk-edge correspondence in the non-Hermitian Su-Schrieffer-Heeger model with the values of the parameters as t1 = 0, t2 = 1, and t3 = 1/5. a Phase diagram on the γ1 -γ2 plane. The blue and the white regions represent the topological insulator (TI) phase with the winding number w being 1 and the normal insulator (NI) phase with w = 0, respectively. The orange region represents a gapless phase. b Energy levels in a finite open chain calculated along the green arrow in a with γ2 = 1.4. Note that γc  1.89. The topological edge states are shown in red. c,d Loops + (red) and − (blue) on the R plane. The values of the parameters are c γ1 = −1 and γ2 = 1.4, and d γ1 = 2.1 and γ2 = 1.4. Note that − passes the origin in d, which corresponds to exceptional points

− passes the origin on the R plane as shown in Fig. 4.7d, where the system has exceptional points. We note that the winding number is not well defined in this case.

4.4 Summary In this chapter, we establish the non-Bloch band theory in a 1D non-Hermitian tight-binding system. We explain how to construct the GBZ, which is given by the trajectories of β M and β M+1 satisfying the condition |β M | = |β M+1 | and show that the Bloch wave number becomes complex in an open chain in general. By applying this theory to some models, we show some examples for the calculations of the GBZ and the energy spectra. In particular, in the non-Hermitian SSH model, we reproduce the non-Bloch band theory by deriving the boundary equation which the eigenstates of the real-space Hamiltonian satisfy. In addition, in this model, the bulk-edge correspondence between the winding number defined by the GBZ and the existence of the topological edge states can be established. Here, we comment on the limit of L → ∞. In Hermitian tight-binding models, as the number of unit cells increases while keeping the lattice constant unchanged, the dispersions of eigenenergies in a large periodic chain asymptotically approach those in a large open chain. This is why eigenenergies and eigenstates calculated by the conventional Bloch band theory well describe these for real materials having open boundaries. In contrast, we have shown in Chap. 3 that this common sense in Hermitian systems is no longer true in non-Hermitian systems. In this sense, the limit of L → ∞ in the discussion in this chapter is not meant to be a mathematically rigorous one. We study the non-Bloch band theory of nonHermitian systems as the natural extension of the Bloch band theory of Hermitian systems. Namely, we only investigate the asymptotic behavior of the distribution of energy levels in a non-Hermitian tight-binding model with a large system size,

4.4 Summary

55

similar to Hermitian cases. Hence, from the viewpoint of mathematics, the results obtained here is just conjecture, and a mathematically rigorous proof of Eq. (4.6) is left as a future work. Nevertheless, our main result, Eq. (4.6), well describes fundamental physics, such as energy spectra, in various non-Hermitian models. In fact, one can show the bulkedge correspondence in non-Hermitian systems in terms of the non-Bloch band theory. Therefore, we conclude that the results obtained in this chapter are physically reasonable and are justified in some perspectives.

Appendix In this Appendix, we prove that the GBZ becomes a unit circle in Hermitian systems. In the following, we assume that the characteristic Eq. (4.3) is an algebraic equation for β of 2Mth degree. In this case, the characteristic equation can be written as M 

ai β i = 0,

(4.68)

i=−M

where ai (i = −M, . . . , M) are coefficients satisfying a−i = ai∗ , and a0 is real. These coefficients are functions of the eigenenergy E and the hopping terms ti,μν included in Eq. (4.1). Here, the eigenenergy is real due to Hermiticity of the Hamiltonian. Equation (4.68) has 2M solutions, and they are numbered so as to satisfy Eq. (4.5). Since we can rewrite Eq. (4.68) as M  i=−M

M   −i  −i = ai β ∗ = 0, (a−i )∗ β ∗

(4.69)

i=−M

the solutions always appear in pairs: (β, 1/β ∗ ), and we get β2M+1− j = 1/β ∗j . In ∗ . Therefore, particular, the relationship between β M and β M+1 leads to β M+1 = 1/β M Eq. (4.6) can be rewritten as |β M | = |β M+1 | = 1.

(4.70)

Thus, we conclude that in general, the GBZ becomes a unit circle in Hermitian systems.

References 1. Yao S, Wang Z (2018) Phys Rev Lett 121:086803 2. Lee CH, Thomale R (2019) Phys Rev B 99:201103(R)

56 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

4 Non-Bloch Band Theory of Non-Hermitian Systems … Yokomizo K, Murakami S (2019) Phys Rev Lett 123:066404 Yokomizo K, Murakami S (2020) Prog Theor Exp Phys 2020 Song F, Yao S, Wang Z (2019) Phys Rev Lett 123:246801 Zhang K, Yang Z, Fang C (2020) Phys Rev Lett 125:126402 Yang Z, Zhang K, Fang C, Hu J (2020) Phys Rev Lett 125:226402 Lee TE (2016) Phys Rev Lett 116:133903 Brody DC (2104) J Phys A 47:035305 Ryu S, Schnyder AP, Furusaki A, Ludwig AW (2010) New J Phys 12:065010 Yokomizo K, Murakami S (2020) Phys Rev Res 2:043045 Ryu S, Hatsugai Y (2002) Phys Rev Lett 89:077002

Chapter 5

Topological Semimetal Phase with Exceptional Points in One-dimensional Non-Hermitian Systems

5.1 Non-Hermitian Su-Schrieffer-Heeger Model Before going to discuss a general theory for a topological semimetal (TSM) phase with exceptional points, we specifically study some properties of the energy spectra in the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Throughout the investigation of this model, we reveal how the TSM phase appears and how each energy band is divided. Furthermore, we can relate the topological phase transition with the creation and annihilation of the exceptional points.

5.1.1 Reinvestigation of the Energy Spectra Similar to Chap. 4, we introduce the non-Hermitian SSH model. Throughout this chapter, for convenience, we assume that the real-space Hamiltonian of this system is written as  † † + † t1+ cn,A cn+1,B + t0+ cn,A cn,B + t−1 cn+1,A cn,B H= n

+

† t1− cn,B cn+1,A

 † − † + t0− cn,B cn,A + t−1 cn+1,B cn,A ,

(5.1)

where all the parameters are set to be real. Then, the system preserves the sublattice symmetry (SLS) and the time-reversal symmetry (TRS), which are defined as  H  −1 = −H and T H ∗ T −1 = H , respectively. Here,  and T are unitary matrices satisfying  2 = T T ∗ = +1 [1]. Henceforth, we set the values of the parameters ± = t−1 ∓ −1 . We show the energy eigenvalues as t0± = −t0 , t1± = t1 ± 1 , and t−1 of this system in a periodic chain and in a finite open chain in Fig. 5.1a-d, showing a qualitative difference between these two chains. In particular, in a long open chain, the gapless phase extends over a certain region in the parameter t0 as shown © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_5

57

58

5 Topological Semimetal Phase with Exceptional Points …

Fig. 5.1 a-d Eigenenergies of the non-Hermitian Su-Schrieffer-Heeger model in a finite open chain (red) and in an infinite periodic chain (black), respectively. The values of the parameters are t1 = 1.2, t−1 = 0.5, 1 = 0.3, and −1 = −0.7. e Energy gap in the system with a periodic boundary condition (black) and with an open boundary condition (red). The plot shows their absolute values |E i | (i = 1, 2). Because of the sublattice symmetry, the two energy eigenvalues E 1 and E 2 are opposite in sign, E 1 = −E 2

in Fig. 5.1e. We note that the gap of this system closes at E = 0 because the two eigenenergies E 1 , E 2 satisfy E 1 = −E 2 due to the SLS. Thus, while qualitative difference of the energy spectra between an open chain and a periodic chain have been discussed in various non-Hermitian systems, the appearance of the stable gapless phase only in an open chain is unexpected in a one-dimensional (1D) system.

5.1.2 Mechanism for the Appearance of the Gapless Phase In the previous subsection, we find that the gapless phase appears in the nonHermitian SSH model. In this subsection, we study the appearance of the gapless phase as shown in Fig. 5.1e in terms of the non-Bloch band theory. First of all, we review the condition for the generalized Brillouin zone (GBZ). By the procedure explained in Sect. 4.1.1, the non-Bloch matrix of the non-Hermitian SSH model is expressed as the off-diagonal form  H (β) = with

0 R+ (β) R− (β) 0

 (5.2)

5.1 Non-Hermitian Su-Schrieffer-Heeger Model

R± (β) = (t1 ± 1 ) β − t0 + (t−1 ∓ −1 ) β −1 ,

59

(5.3)

where β ≡ eik , k ∈ C. Then, one can get the characteristic equation as   0 = det H (β) − E = E 2 − R+ (β) R− (β) ,

(5.4)

leading to the eigenenergies E ± (β) with E − (β) = −E + (β). Since Eq. (5.4) is a quartic equation for β, the condition for the GBZ is given by |β2 | = |β3 |

(5.5)

for four solutions of Eq. (5.4) satisfying |β1 | ≤ |β2 | ≤ |β3 | ≤ |β4 | ,

(5.6)

and the trajectories of β2 and β3 give the GBZ. Now, we explain the mechanism for the appearance of the gapless phase. In our model, the solutions of the equations R± (β) = 0 are gap-closing points, shown as the red and blue dots and squares in Fig. 5.2c-1-h-1. These gap-closing points become

Fig. 5.2 a Phase diagram in the non-Hermitian Su-Schrieffer-Heeger model for a long open chain with the parameter values as t1 = 1.2, t−1 = 0.5, and 1 = 0.3. At the red star (t0 = −1.5922 and −1 = 0.2), the gap closes, and the direct transition between two insulator phases with w = 0 (normal insulator (NI), white region) and w = 1 (topological insulator (TI), blue region) occurs. Here, w is the winding number given in Eq. (5.8). The orange regions express the topological semimetal (TSM) phase. b Band gap along the black arrow (−1 = −0.7) in (a). Since the system is a two-band model with the sublattice symmetry, the two eigenenergies satisfy E + (β) = −E − (β). Hence, we only show |E| to see whether the gap closes. c-h Gap-closing points, generalized Brillouin zone, and motion of the exceptional points (red star) along the black arrow in (a). The red (blue) dots and squares express the gap-closing points of the equation R+ (β) = 0 (R− (β) = 0)

60

5 Topological Semimetal Phase with Exceptional Points …

a gapless points when the GBZ goes through them. Let a β = βia (i = 1, 2, a = +, −) denote the gap-closing points of Ra (β) = 0, with β1 ≤ β2 . In the regions A and B in Fig. 5.2a, + − − + β ≤ β = β ≤ β (5.7) 1 1 2 2



∗ holds. Here, the equality β1− = β2− follows from β1− = β2− because β1− and − β2 are solutions of an algebraic equation with real coefficients. Thus, the condition − − (5.5) is satisfied; therefore, β1− and β2− are on the ∗ Then, β1 and β2 are the

−GBZ. − gapless points. Importantly, the condition β1 = β2 remains satisfied even when the values of the system parameters change. Namely, the gapless points move along the GBZ as shown in Fig. 5.2c-2-h-2, and the system remains in the gapless phase as shown in Fig. 5.2b. From another perspective, as the values of the system parameter change continuously, the GBZ is deformed so that the gapless points always lie on the GBZ. We note that in the regions C and D in Fig. 5.2a, a similar scenario holds true by exchanging R+ (β) and R− (β).

5.1.3 Winding Number, Exceptional Points, and Topological Phase Transition In this subsection, we discuss the topological aspects of the gapless phase shown in Sect. 5.1.2. To this end, in the non-Hermitian SSH model, we introduce the winding number and classify the topological phase in this system. The winding number is given by  1  w+ − w− , w± = arg R± (β) Cβ , (5.8) w=− 2 2π   where arg R± (β) Cβ means the change of the phase of the functions R± (β) as β goes along the GBZ denoted as Cβ in a counterclockwise way. As long as there is a gap at E = 0, R± (β) never vanish along the GBZ, and the winding number is well defined. Here, the gapped phase is defined as E + (β) = E − (β) for every β on the GBZ. In fact, Fig. 5.2a shows the phase diagram in the non-Hermitian SSH model, with the normal insulator (NI) phase with w = 0 (white region), the topological insulator (TI) phase with w = 1 (blue region), and the gapless phase (orange region). Importantly, we find that this gapless phase always appears as an intermediate phase between the NI and TI phases. Therefore, the gapless phase is the TSM phase. Furthermore, in the TSM phase, the gap closes at E = 0, meaning that the equation R+ (β) = 0 or R− (β) = 0 holds somewhere on the GBZ. At such a point on the GBZ, since the Hamiltonian cannot be diagonalizable, this point is an exceptional point. Namely, in the TSM phase, the system has the exceptional points. In the following, in order to study the role of the TSM phase in the topological phase transition, we relate the creation and annihilation of the exceptional points with the change of the value of the winding number. First of all, we focus on the

5.1 Non-Hermitian Su-Schrieffer-Heeger Model

61

motion of the gap-closing points of the equations R± (β) = 0. Along the black arrow in Fig. 5.2a, we show the position of the gap-closing points β1+ and β2+ of R+ (β) = 0 (β1− and β2− of R− (β) = 0) as the red (blue) dots and squares, respectively, as shown in Fig. 5.2c-1-h-1. When we decrease the value of the parameter t0 , at t0 = 2.683 [Fig. 5.2g-1], β1+ and β2+ change from real values to complex values via coalescence. This corresponds to a pair creation of the exceptional points. After the coalescence,  + + ∗ with β = β , and their common absolute value is β1+ and β2+ become complex, 2 1 + that β and β2+ stay on between the values of β1− and β2− , meaning 1 +the At − GBZ.  β = β + . of t0 = 0.5813 [Fig. 5.2d-1], the value of β2 becomes equal to that 1 2 − − + After passing this point (i.e., t0 becomes less than 0.5813), β1 < β2 < β1 = β + , meaning that β + and β + are no longer on the GBZ, and the gap opens 2 1 2 [Fig. 5.2c-1]. At the phase transition point [Fig. 5.2d-1], three solutions β1+ , β2+ , β2− share the same absolute value. In this case, the gap closes at the cusps on the GBZ. The change of the value of the winding number readily follows from the following argument. In the NI phase, the GBZ surrounds one solution of R+ (β) = 0 of R− (β) = 0 [Fig. 5.2h-1]. In this case, in Eq. (5.8), w± = and one solution  arg R± (β) Cβ /2π = 0 because the functions R± (β) are proportional to 



β − β1± β − β2± /β. On the other hand, in the TI phase, there exist two solutions of R− (β) = 0 and no solutions of R+ (β) = 0 inside the GBZ [Fig. 5.2c-1], which leads to w− = 1, w+ = −1, and the winding number takes 1 as expected. Therefore, the creation and annihilation of the exceptional points changes the number of the solutions of R± (β) = 0 inside the GBZ. As a result, the value of the topological invariant also changes. Namely, the value of the winding number changes through the motion of the exceptional points on the GBZ. In conclusion, this TSM phase appears as an intermediate phase between the NI and TI phases. We will discuss this point in general cases in Sect. 5.2. We note that in addition to the topological phase transition between two insulator phases via the TSM phase, direct phase transition from the NI phase to the TI phase is also possible as shown in the inset of Fig. 5.2a. In this case, both R+ (β) and R− (β) simultaneously become zero.

5.1.4 TRS-Unbroken Region, STS-Unbroken Region, and TRS/STS-Broken Region In this subsection, we discuss the structure of the energy spectra and the stabilization of the exceptional point from another perspective. Within a single band, the energy eigenvalues change along either the real or the imaginary axis to some extent, and then, they abruptly split off the axes. We find that this behavior comes from a unique and remarkable property that the single band is divided into three regions as shown in Fig. 5.3. In the first region,  the energies are real, and the eigenenergies of a timereversal pair |ψ , T |ψ∗ are degenerate, i.e., E = E ∗ , where |ψ is an eigenstate of the Hamiltonian (5.1). In the second region, the energies are pure imaginary, and

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5 Topological Semimetal Phase with Exceptional Points …

Fig. 5.3 Energy spectra and generalized Brillouin zone (GBZ) in the non-Hermitian Su-SchriefferHeeger model. The TRS-unbroken region, the STS-unbroken region, and the TRS/STS-broken region are shown in green, blue, and orange, respectively. The values of the parameters are t1 = 1.2, t−1 = 0.5, 1 = 0.3, and −1 = −0.7, with a t0 = 0.4 and b t0 = 1.1. Ai and Bi (i = 1, . . . , 4) are the cusps on the GBZ

a pair of states |ψ and T |ψ∗ related by the sublattice-time-reversal symmetry (STS) is degenerate, i.e., E = −E ∗ . In the third region, the energies are complex, and neither the time-reversal pair nor the sublattice-time-reversal pair is degenerated. We call these three regions TRS-unbroken region, STS-unbroken region, and TRS/STS-broken region, respectively. These regions are connected to each other at the cusps or the exceptional points. In Fig. 5.3a-1 and b-1, three curves meet at one point, where the GBZ has the cusps represented by Ai and Bi (i = 1, . . . , 4). It comes from the property that three points on the GBZ share the same energy. Furthermore, in Fig. 5.3b, the green and blue lines are connected at the exceptional point with E = 0. Thus, the exceptional point connecting the real and pure imaginary energies becomes stable because such structure is topologically protected by the symmetries.

5.2 Topological Semimetal Phase with Exceptional Points In this section, we study the TSM phase with exceptional points in general cases. We focus on a 1D non-Hermitian tight-binding system with the SLS and the TRS and explain the mechanism for the appearance of the TSM phase in this system. Finally, we comment on the behavior of the gapless phase when we break either the SLS or the TRS.

5.2 Topological Semimetal Phase with Exceptional Points

63

5.2.1 Concept First of all, we show that the TSM phase appears in a 1D non-Hermitian tight-binding system with the SLS and the TRS. Due to the SLS, one can write the non-Bloch matrix of this system as an off-diagonal form, again explicitly written as   0 R+ (β) H (β) = , (5.9) R− (β) 0 where β = eik , k ∈ C, and R± (β) are N × N matrices when the system has 2N bands. Then, the characteristic equation   0 = det H (β) − E   (5.10) = det R+ (β) R− (β) − E 2 yields the bands symmetric with respect to E = 0. Furthermore, the condition for the GBZ is given by |β M | = |β M+1 | (5.11) for the solutions β1 , . . . , β2M (|β1 | ≤ · · · ≤ |β2M |) of the characteristic equation, being an algebraic equation for β with an even degree 2M. We note that the trajectories of β M and β M+1 give the GBZ. Now, a condition for a gap closing at E = 0 is decomposed into two equations as det R+ (β) = 0 and det R− (β) = 0. Thanks to the TRS, det R± (β) are polynomials of β and β −1 with real coefficients. Namely, it follows that any complex solutions of det R± (β) = 0 appear in complex conjugate pairs (β, β ∗ ). Then, when we suppose β M and β M+1 form a pair of the complex conjugate solutions of det R+ (β) = 0 (or det R− (β) = 0), we have Eq. (5.11), meaning that β M and β M+1 are on the GBZ, and the gap closes. Therefore, even when the values of the system parameters change, the gap remains zero as long as this pair gives Mth and (M + 1)th largest absolute values among the 2M solutions. In Sect. 5.1.3, we discuss how the creation and annihilation of the exceptional points occur in the non-Hermitian SSH model. We can generalize the motion of the exceptional points on the GBZ. Namely, because of the mechanism for the stabilization of the TSM phase, annihilations (and likewise creations) of the exceptional points are limited to two patterns as shown in Fig. 5.4a and b. Figure 5.4a represents a coalescence of two exceptional points, and Fig. 5.4b represents an encounter between the gap-closing point, which is the solution of the equation det R+ (β) = 0 (or det R− (β) = 0), and the cusp. In Fig. 5.4a, the two exceptional points meet, become two real gap-closing points, and move away from the GBZ. As an example of the non-Hermitian SSH model, this can be seen in Fig. 5.2g. On the other hand, the case of Fig. 5.4b occurs when two complex-conjugate exceptional points and one gap-closing point share the same absolute value. If it occurs, for example like Fig. 5.2d, the gap closes at the three points on the GBZ. As we change the system parameters, the ordering of the absolute values of three gap-closing points changes, allowing the exceptional point to disappear and the gap to open.

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5 Topological Semimetal Phase with Exceptional Points …

(a)

(b)

Im(β)

Im(β)

Re(β)

Cβ

Re(β)

Cβ

Fig. 5.4 a Coalescence of the exceptional points. b Annihilation of the exceptional point at the cusp. The yellow and blue dots express the gap-closing points and the exceptional points, respectively. The red stars are the cusps of the generalized Brillouin zone denoted as Cβ

5.2.2 Winding Number and Creation and Annihilation of the Exceptional Points In this subsection, we show that in a 1D non-Hermitian system with both the SLS and the TRS, the value of the winding number changes through the creation and annihilation of the exceptional points as shown in Fig. 5.4. First of all, we focus on a two-band model. The non-Bloch matrix H (β) of this system is given in Eq. (5.9) with N = 1. In the following, without loss of generality, we can write holomorphic functions R± (β) as 2M  C±

β − βi± , (5.12) R± (β) = M β i=1 where C± are real constants due to the TRS. Then, the characteristic equation can be explicitly written as 2M 

 C+ C−

β − βi+ β − βi− = E 2 , 2M β i=1

(5.13)

which is an algebraic equation for β with a degree 4M. Here, by numbering the solutions of Eq. (5.13) so as to satisfy |β1 | ≤ · · · ≤ |β4M |, the condition for the GBZ can be written as |β2M | = |β2M+1 | , (5.14) and one can get the GBZ from the trajectories of β2M and β2M+1 . As mentioned in Chap. 4, the GBZ always encircles the origin on the complex plane [2, 3]. This system is classified in terms of the winding number defined in Eq. (5.8). We note that as long as there is a gap at E = 0, R± (β) never vanish along the GBZ, and the winding number is well defined. On the other hand, when the gap closes at E = 0, the gap-closing condition can be decoupled into two independent equations as R+ (β) = 0 and R− (β) = 0 being satisfied somewhere on the GBZ. At such a point, since the non-Bloch matrix cannot be diagonalizable, this point is an exceptional point, and in this case, the winding number is not well defined.

5.2 Topological Semimetal Phase with Exceptional Points

65

From Eq. (5.12), we can rewrite the form of the winding number as w=−

+ − − Nzeros Nzeros , 2

(5.15)

± where Nzeros expresses the number of solutions of R± (β) = 0 inside the GBZ, respectively. Furthermore, it is worth noting that from Ref. [2], when the system has a gap around E = 0, we can get

1 2π

d log det H (β) = Nzeros − 2M = 0,

(5.16)

Cβ

 + − where Nzeros = Nzero expresses the number of solutions of the equation + Nzero det H (β) = 0 inside the GBZ denoted as Cβ . Equation (5.16) tells us that the total number of solutions of det H (β) = 0 inside the GBZ, namely the number of gapclosing points inside the GBZ, is unchanged as long as the system has a gap. Now, we focus on two insulator phases separated by the TSM phase. When the system enters the TSM phase from one of the insulator phases, the exceptional points are created by the inverse process as shown in Fig. 5.4a (or Fig. 5.4b). Then, after the further change of system parameters, the system becomes the other insulator phase from the TSM phase, and here, the exceptional points are annihilated by the process as shown in Fig. 5.4b (or Fig. 5.4a). If the creation is by the inverse process of Fig. 5.4a and the annihilation is by the

 process of Fig. 5.4b (or vice versa), while the + − + Nzero of solutions of det H (β) = 0 inside the GBZ total number Nzeros = Nzero + and is unchanged, such a motion of the exceptional points change the value of Nzeros − Nzeros by 1 and −1 (or by −1 and 1), respectively, resulting in the change in the value of the winding number by 1 (or by −1). Therefore, we conclude that these insulator phases have different values of the winding number. We comment that this scenario can be extended to a multi-band system if all energy bands have common GBZ, and the winding number can be defined in Eq. (4.56). On the other hand, when each GBZ is associated with each energy band, it is difficult to modify the above proof because the definition of the winding number in such a case has not been revealed yet.

5.2.3 Symmetry-Breaking Effect Finally, we discuss the effect of symmetry breaking. When the SLS is broken while the TRS is preserved, the gapless phase survives because of the reality of the coefficients of the characteristic Eq. (5.10), but the gap closes not necessarily at E = 0. On the other hand, when the TRS is broken, either the SLS or the pseudo-particle-hole symmetry (PHS) [1], defined in Eq. (5.17), is preserved. The TSM phase disappears because the coefficients of Eq. (5.10) become complex. In conclusion, the TSM phase in a 1D non-Hermitian system is robust against the change of the system parameters, provided both the SLS and the TRS are preserved.

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5 Topological Semimetal Phase with Exceptional Points …

5.3 Instability of the Gapless Phase in a 1D Non-Hermitian System with Pseudo-Particle-Hole Symmetry In this section, we study the symmetry-breaking effect for the gapless phase that emerged in the non-Hermitian SSH model. To this end, first, we investigate a 1D non-Hermitian system with the pseudo-PHS and the topological classification in this system. We show that although the gapless phase also appears in this system, this phase is unstable against any perturbations preserving the pseudo-PHS.

5.3.1 Pseudo-Particle-Hole Symmetry In the following, we focus on a two-band model with the pseudo-PHS. Here, for a real-space Hamiltonian H , this symmetry is defined as CH ∗ C−1 = −H,

(5.17)

where C is the unitary matrix, and CC∗ = +1. We note that this symmetry is the product between the SLS and the TRS. In this system, the non-Bloch matrix can be written as  H (β) = H0 (β) σ0 + Hi (β) σi , (5.18) i=x,y,z

where σ0 is a 2 × 2 identity matrix, σi (i = x, y, z) are the Pauli matrices, and β ≡ eik , k ∈ C. We assume that it satisfies  ∗

 σx H (β) σx−1 = −H β ∗ , (5.19) and then, we have 

Hi (β)

∗

 

 ∗ = −Hi β ∗ (i = 0, x, y) , Hz (β) = Hz β ∗ .

(5.20)

It is worth noting that when arg β = 0 and arg β = π , the functions Hi (β) (i = 0, x, y) are pure imaginary and the function Hz (β) is real.

5.3.2 Z2 Topological Invariant In this subsection, we show that a 1D non-Hermitian system with the pseudo-PHS is classified in terms of a Z2 topological invariant. For the non-Bloch matrix (5.18), we can define it as

βπ  d  1 arg R+ (β) − arg R− (β) (mod 2) , dβ ν= 2π β0 dβ

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67

Fig. 5.5 Generalized Brillouin zone in the generalized non-Hermitian Su-Schrieffer-Heeger model. The values of the parameters are t = t = 0, t1 = 1.2, t0 = 1, and t−1 = 0.5; a 1 = 0.3, −1 = 0.8, and w1 = w−1 = 0, and b 1 = 0.3, −1 = 0.8, w1 = 0.5, and w−1 = 0.8. The points where the GBZ intersects the positive and negative sides of a real axis are denoted by β0 and βπ , respectively

 R± (β) = Hz (β) ± i Hx2 (β) + H y2 (β),

(5.21)

where β0 (βπ ) is the value of β at arg β = 0 (arg β = π ) on the GBZ (i.e., see Fig. 5.5). In Eq. (5.21), the integral contour β goes along the GBZ, and we select the branch cut of the square root so that the both functions R± (β) becomes continuous on the GBZ. In the following, we assume that a system has a gap. Here, we note that two energy bands are separated by a line which determines a complex gap on the complex energy plane. This gap is called line gap [1]. Now, in order to show that ν takes only 0 or 1, we calculate the value of exp (2πiν). Since the two functions R± (β0 ) and R± (βπ ) take real values, we can rewrite the expression of exp (2πiν) as       exp (2πiν) = exp i arg R+ (βπ ) − arg R+ (β0 ) exp i arg R− (βπ ) − arg R− (β0 )     sgn R+ (βπ ) sgn R− (βπ )     = sgn R+ (β0 ) sgn R− (β0 ) sgn [Rσ (β)] = β=β0 ,βπ σ =±

=

β=β0 ,βπ

⎡

sgn ⎣



i=x,y,z

⎤ Hi2 (β)⎦ .

(5.22)

  We note that the quantities i Hi2 (β0 ) and i Hi2 (βπ ) are real.  Since we assume the presence of the line gap, we conclude that i Hi2 (β0 ) and i Hi2 (βπ ) have the same sign, as we prove by contradiction in the following.

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5 Topological Semimetal Phase with Exceptional Points …

Suppose

 i

Hi2 (β0 ) and 

 i

Hi2 (βπ ) have different signs. We can set

Hi2 (β0 ) > 0,

i=x,y,z



Hi2 (βπ ) < 0

(5.23)

i=x,y,z

without loss of generality. First of all, we assume H0 (β) = 0 for simplicity. At  2 β = β0 , the energies are E = ±ε0 , ε0 = i Hi (β0 ) > 0. Now, we choose E =  2 ε0 = i Hi (β0 ), and we change β along the GBZ in a counterclockwise way from β = β0 to β = βπ . Here, let C+ denote this path on the complex plane. Then,  2 at β = βπ , the energy is given by E = επ = i Hi (βπ ) ∈ iR, where the branch  2 of the square root is chosen in such a way that E = i Hi (β) is continuous along C+ . Next, we consider GBZ β0 − along  a 2pathC  the  in a clockwise way from β = ∗ 2 ∗ ∗ to β = βπ . Since = i Hi (β) i Hi (β ) , the energy at β and that at β are complex-conjugate. Namely, (C+ )∗ = C− , and hence, the energy at β = βπ along C− is E = επ∗ = −επ . Thus, by encircling the GBZ (= −C− + C+ ) once from βπ to βπ , the branch changes from E = −επ to E = επ , meaning that the two energies  2 E =± i Hi (β) are continuously connected. Therefore, there is no line gap,   contradicting the assumption. Thus, we conclude that i Hi2 (β0 ) and i Hi2 (βπ ) have the same sign. So far, we assume H0 (β) = 0, but even in the case of H0 (β) = 0, since this term does not affect the above argument, the above proof remains valid. From Eq. (5.21), we can get exp (2πiν) = 1, and we conclude that the value of ν is an integer. As a result, ν can take only 0 or 1 (mod 2). In conclusion, we can interpret ν as the Z2 topological invariant in this system. In Hermitian cases, we can greatly simplify the formula of the Z2 topological invariant. The Bloch wave number k becomes real, and the functions Hi (β) (i = 0, x, y, z) become real functions. In the following, we replace Hi (β) by Hi (k) (i = 0, x, y, z) , k ∈ [−π, π ]. In this case, the conventional PHS produces that Hi (k) = −Hi (−k) (i = 0, x, y) , Hz (k) = Hz (−k)

(5.24)

and that Hi (k) (i = 0, x, y) become zero at k = 0 and k = π . Then, we can get       arg Hz − i Hx2 + Hy2 = − arg Hz + i Hx2 + Hy2

(5.25)

and can rewrite Eq. (5.21) as

   1 π d ν= dk arg Hz (k) + i Hx2 (k) + Hy2 (k) π 0 dk  π 1 arg Hz (k) 0 . = π

(5.26)

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69

Since Eq. (5.24) tells us that both argHz (0) and argHz (π ) take 0 or π (mod 2π ), Eq. (5.26) can be further rewritten as  ν=

    0 if sgn  Hz (0) sgn  Hz (π ) > 0, 1 if sgn Hz (0) sgn Hz (π ) < 0.

(5.27)

At last, we obtain the known formula [4]     (−1)ν = sgn Hz (0) sgn Hz (π ) .

(5.28)

5.3.3 Generalized Non-Hermitian Su-Schrieffer-Heeger Model In this subsection, we study the generalized non-Hermitian SSH model. The realspace Hamiltonian of this system can be written as  † † A † H= it1A cn,A cn+1,A + it0A cn,A cn,A + it−1 cn+1,A cn,A n † B † B † +it1 cn,B cn+1,B + it0B cn,B cn,B + it−1 cn+1,B cn,B + † + † + † +t1 cn,A cn+1,B + t0 cn,A cn,B + t−1 cn+1,A cn,B

 † † − † + t1− cn,B cn+1,A + t0− cn,B cn,A + t−1 cn+1,B cn,A ,

(5.29)

where all the parameters are set to be real, meaning that the real-space Hamiltonian (5.29) satisfies the pseudo-PHS. Henceforth, we set all the parameters as t0A = t0B = ± A = t ∓ w±1 , t±B = t ± w±1 , t0± = −t0 , t1± = t1 ± 1 , and t−1 = t−1 ∓ −1 . t, t±1 The non-Bloch matrix can be written in the form of Eq. (5.18), and the functions Hi (β) (i = 0, x, y, z) are given by ⎧

 H0 (β) = it + it β + β −1 , ⎪ ⎪ ⎨ Hx (β) = t1 β − t0 + t−1 β −1 , −1 H y (β) = i  ⎪ ⎪

1 β − −1 β −1 , ⎩ Hz (β) = −i w1 β − w−1 β .

(5.30)

We note that the non-Bloch matrix satisfies the condition for the pseudo-PHS given in Eq. (5.19) with the replacement x → y, y → z, and z → x. As we discussed previously, the system can be classified in terms of the Z2 topological invariant. In this case, the form of Eq. (5.21) is obtained by replacing the variables; x → y, y → z, and z → x.   Since the characteristic equation det H (β) − E = 0 is a quartic equation for β, the condition for the GBZ can be written as |β2 | = |β3 | when the solutions satisfy |β1 | ≤ |β2 | ≤ |β3 | ≤ |β4 |. The examples of the GBZ and the energy spectra are given in Figs. 5.5 and 5.6a-3, respectively.

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5 Topological Semimetal Phase with Exceptional Points …

Fig. 5.6 Phase diagram and bulk-edge correspondence in the generalized non-Hermitian SuSchrieffer-Heeger model. a Phase diagram with t = t = w−1 = 0, t1 = 1.2, t−1 = 0.5, 1 = 0.3, and w1 = 0.2. We note that in this case, H0 (β) = 0. a-1 Phase diagram on the t0 -−1 plane. The blue region represents the topological insulator phase with the Z2 topological invariant ν being 1, the white region represents the normal insulator phase with ν = 0, and the orange region represents the semimetal phase with exceptional points. Along the black arrow in a-1 with −1 = 0.2, we show the results for a-2 energy levels in a finite open chain and a-3 the energy spectra. The edge states are shown in red in (a-2). a-4 Shows + (red) and − (blue) on the R plane with t0 = 1 and −1 = 0.2. These loops encircle the origin on the complex plane, meaning that the value of Z2 is equal to 1. b Phase diagram with H0 (β) = 0. The parameters are set as t = w−1 = 0, t1 = 1.2, t−1 = 0.5, 1 = 0.3, and w1 = 0.2, and the parameter t takes infinitesimal values

Let ± denote the loops drawn by the functions R± (β) given in Eq. (5.21) on the complex plane when β goes along the GBZ in a counterclockwise way. Equation (5.21) tells us how to determine the value of the Z2 topological invariant; when neither + nor − surrounds the origin on the complex plane, it is equal to 0, and when two loops simultaneously surround the origin, it is equal to 1. For example, in

5.3 Instability of the Gapless Phase in a 1D Non-Hermitian System …

71

the case of Fig. 5.6a-4, the Z2 topological invariant takes 1. We note that the system has exceptional points when either + or − passes the origin, and the Z2 topological invariant is not well defined. We can get the phase diagram in the generalized non-Hermitian SSH model as shown in Fig. 5.6a-1 and can confirm that the topological edge states appear when the Z2 takes the nonzero value as shown in Fig. 5.6a-2. Therefore we can establish the bulk-edge correspondence between the Z2 topological invariant and the existence of the topological edge states. In this model, we can see that the pseudo-PHS cannot topologically protect the semimetal phase shown in orange in Fig. 5.6a. For simplicity, let the value of the parameter t be zero. So far, we have set t = t = 0 in the calculations in Fig. 5.6a-1a-4. On the other hand, for t = 0 being an infinitesimal value as an example, we can obtain the phase diagram as shown in Fig. 5.6b. We can confirm that the semimetal phase in Fig. 5.6a-1 disappears by putting t = 0. We note that the exceptional points appear on the orange lines on the phase diagram. However, this exceptional point can be removed by adding other perturbation terms. Therefore, we conclude that the pseudo-PHS cannot topologically protect the semimetal phase.

5.4 Summary In summary, the appearance of the TSM phase with exceptional points is attributed to the unique features of the GBZ. It is shown that in a 1D non-Hermitian system with the SLS and the TRS, the TSM phase is stabilized, unlike Hermitian systems. We also find that each energy band is divided into three regions in terms of the symmetry of the eigenstates, and the regions switch only at the cusps and the exceptional points on the GBZ. We here emphasize that the TSM phase is topologically protected not by the pseudo-PHS but by both the SLS and the TRS.

References 1. 2. 3. 4.

Kawabata K, Shiozaki K, Ueda M, Sato M (2019) Phys. Rev. X 9:041015 Zhang K, Yang Z, Fang C (2020) Phys. Rev. Lett. 125:126402 Yang Z, Zhang K, Fang C, Hu J (2020) Phys. Rev. Lett. 125:226402 Kitaev AY (2001) Physics-Uspekhi 44:131

Chapter 6

Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

6.1 Bosonic Bogoliubov-de Gennes Systems In this section, we study the non-Bloch band theory in bosonic systems described by a Bogoliubov-de Gennes (BdG) Hamiltonian. To this end, we review the formulation of the eigenvalue problem of the bosonic BdG Hamiltonian in real space.

6.1.1 Real-Space Bogoliubov-de Gennes Hamiltonian First of all, we review the formulation of the eigenvalue problem of the bosonic BdG Hamiltonian in real space. We start with a one-dimensional (1D) bosonic tightbinding system. The Hamiltonian of this system is written as   1 †  a , (6.1) a a HBdG H= a† 2   where a = . . . , a1,1 , . . . , a1,q , . . . , a L ,1 , . . . , a L ,q , . . . . We note that a unit cell is composed of q degrees of freedom, and a j,σ (σ = 1, . . . , q) represents a bosonic annihilation operator at the jth unit cell. Then, the BdG Hamiltonian preserves the particle-hole symmetry (PHS) [1] defined as T τx−1 = HBdG τx HBdG

(6.2)

because    a† 1  †  −1 a a τx HBdG τx H = a 2   T a 1  †  1 = − Tr (τz HBdG ) a a τx HBdG τx−1 a† 2 2 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_6

73

74

6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

=

 1  †  T a a τx HBdG τx−1 2



a a†



1 − Tr (τz HBdG ) 2

(6.3)

should be equal to Eq. (6.1). Here, τx and τz are defined as  τx =

O 1l 1l O



 , τz =

 1l O , O −1l

(6.4)

respectively, where O and 1l are a zero matrix and an identity matrix with the dimension being equal to that of the vector a, respectively. Hence, a matrix form of the BdG Hamiltonian is given by [1]  HBdG =

M  † M T

 .

(6.5)

In analogy with a fermionic BdG Hamiltonian representing a Bardeen-CooperSchrieffer superconductor, the Hermitian matrix M represents the normal parts, and the symmetric matrix  does the pairing terms. In order to get the energy eigenvalues of the system, diagonalize the BdG Hamiltonian by the basis transforma we should  tion a† a = α † α T † , where α is another set of bosonic annihilation operators. Here, since a and α satisfy the bosonic commutation relation, T must be a paraunitary matrix [2–6] defined as (6.6) T † τz T = τz . Then, the BdG Hamiltonian (6.1) is diagonalized as T −1 (τz HBdG ) T =



  O , O −

(6.7)

where  is a diagonal matrix. As a result, we have 1 †  α α H= 2



O O 



α α†

 .

(6.8)

We note that the columns of T are the right eigenvectors of τz HBdG . On the other hand, since τz HBdG has the pseudo-Hermiticity [1] defined as τz (τz HBdG )† τz = τz HBdG ,

(6.9)

the rows of T † τz are the left eigenvectors of τz HBdG . Hence, if a set of eigenvectors of τz HBdG forms the biorthogonal basis [7], we can get the paraunitary matrix T . In other words, the constitution of the biorthogonal basis is equivalent to the condition (6.6). Now, we can explicitly write the eigenvalue equation of τz HBdG as  (τz HBdG )

U V∗ V U∗



 =

U V∗ V U∗



  O , O −

(6.10)

6.1 Bosonic Bogoliubov-de Gennes Systems

75

where  T =

U V∗ V U∗

 (6.11)

is a paraunitary matrix. This indicates that since τz HBdG is non-Hermitian, some features of non-Hermitian physics may arise, although the original Hamiltonian (6.1) is Hermitian. We note that if the BdG Hamiltonian is positive definite, all the eigenenergies of the system become real, and the system can be regarded as a Hermitian system [1]. On the other hand, if not, the bosonic BdG system is essentially nonHermitian, and the eigenenergies are complex in general. Importantly, in this case, we can get the energy eigenvalue by applying the non-Bloch band theory, as we will discuss later.

6.1.2 Non-Bloch Band Theory Now, we investigate the non-Bloch band theory in a 1D bosonic BdG system. In the following, we assume that the ranges of the hopping in the normal terms M and the pairing terms  are up to Ns and N p unit cells, respectively. Then Eq. (6.10) can be explicitly written as ⎧ ⎡ ⎤ Np q Ns ⎪





⎪ ⎪ ⎪ ⎣ Mi,σ τ u κj+i,τ + i,σ τ v κj+i,τ ⎦ = E κ u κj,σ ⎪ ⎪ ⎨ j τ =1 i=−Ns i=−N p ⎡ ⎤ Np q Ns ⎪





⎪ ⎪ ⎪ ⎣− ∗−i,τ σ u κj+i,τ − M−i,τ σ v κj+i,τ ⎦ = E κ v κj,σ ⎪ ⎪ ⎩ j

τ =1

i=−N p

(6.12)

i=−Ns

for σ = 1, . . . , q, where u κj,σ and v κj,σ are the (( j, σ ) , κ) components of the matrices U and V , respectively, and Mi,σ τ and i,σ τ are the (σ, τ ) components of the matrices M and , respectively, representing the hopping to the (−i)th  nearest unit cell. Here, thanks to spatial periodicity in the bulk, the eigenvectors u κj,σ , v κj,σ can be given by the linear combination as 4N  κ 

 κ,m κ,m  u j,σ , v κj,σ = u σ , vσ (βm ) j (σ = 1, . . . , q) ,

(6.13)

m=1

  where N = q max Ns , N p , and β = βm are the solutions of the characteristic equation   det sz HBdG (β) − E = 0 (6.14) of the non-Bloch BdG matrix

76

6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

⎛

Ns

Np

⎞

Mi,σ τ β i i,σ τ β i ⎟ ⎜ ⎟ ⎜   i=−N p ⎟ ⎜ i=−N HBdG (β) σ τ = ⎜ N p s ⎟, Ns ⎟ ⎜

∗ i i⎠ ⎝ −i,τ σ β M−i,τ σ β i=−N p

(6.15)

i=−Ns

where sz is the 2q × 2q matrix expressed as sz = diag (1l, −1l). We note that Eq. (6.14) is an algebraic equation for β of degree 4N . In terms of the non-Bloch band theory discussed in Chap. 4, the generalized Brillouin zone (GBZ) can be obtained from |β2N | = |β2N +1 | , (6.16) where the solutions of Eq. (6.14) are ordered in the following way: |β1 | ≤ |β2 | ≤ · · · ≤ |β4N −1 | ≤ |β4N | .

(6.17)

Namely, the trajectories of β2N and β2N +1 on the complex plane give the GBZ. In general, |β2N | = |β2N +1 | = 1 in Hermitian systems, leading to the conventional Brillouin zone. As explained in previous chapters, in free fermion systems, the GBZ has various features unique to non-Hermitian systems. We expect that the bosonic BdG system inherits such features of the GBZ. Importantly, Eq. (6.13) indicates that the paraunitary matrix T can be given by j the linear combination  † (βm ) (m = 1, . . . , 4N ). In fact, from the basis  †  ofthe† terms transformation a a = α α T , we can get α † = a† U − aV,

(6.18)

and the quasiparticle-creation operator can be explicitly written as † = ακ,σ



 u κj,σ a †j,σ − v κj,σ a j,σ j

4N  

j † κ,m j = u κ,m σ (βm ) a j,σ − vσ (βm ) a j,σ j

(6.19)

m=1

for σ = 1, . . . , q. Here the coefficients u κ,m and vσκ,m are determined by the open σ boundary conditions given as ακ,σ ( j = 0) = ακ,σ ( j = L + 1) = 0

(6.20)

† satisfy the boson statistics and by the condition that the operators ακ,σ and ακ,σ given as   (6.21) ακ,σ , ακ† ,τ = δκ,κ  δσ,τ .

6.1 Bosonic Bogoliubov-de Gennes Systems

77

Therefore, from Eqs. (6.19)–(6.21), we can get the Bogoliubov transformation diagonalizing the BdG Hamiltonian in terms of the non-Bloch band theory.

6.2 Bosonic Kitaev-Majorana Chain In this section, we investigate the bosonic Kitaev-Majorana chain proposed in Ref. [8]. While the occurrence of the non-Hermitian skin effect in this system was proposed in the previous work, we show that the non-Hermitian skin effect is fragile against infinitesimal perturbations. Furthermore, in a special case, we can derive the analytical representation of the Bogoliubov transformation from Eq. (6.19).

6.2.1 Non-Hermitian Property First of all, we start with the real-space Hamiltonian of the bosonic Kitaev-Majorana chain. It is given by H=

 i  

 † t  iφ † μa j a j + e a j+1 a j + e−iφ a †j a j+1 + a †j+1 a †j − a j a j+1 , 2 2 j

(6.22) where a j is a bosonic annihilation operator at the jth site, and all the parameters are set to be positive real numbers, for simplicity. In particular, when φ = 0 and φ = π/2, we only focus on the case of t >  because the system is dynamically unstable if t <  [8]. This model corresponds to the case of q = N = 1 in Sect. 6.1.2. Remarkably, although the Hamiltonian (6.22) is Hermitian, the system is intrinsically non-Hermitian, leading to non-Hermitian phenomena. We note that the previous work only studied non-Hermitian properties in this model with the case of μ = 0 and φ = π/2. The non-Bloch BdG matrix for Eq. (6.22) is written as       t  it β + β −1 σ y − sin φ β − β −1 σz , HBdG (β) = μ + cos φ β + β −1 σ0 − 2 2 2

(6.23)

where β = eik , k ∈ C, σ0 is a 2 × 2 identity matrix, and σi (i = x, y, z) are the Pauli matrices. As mentioned in Sect. 6.1.2, we can get the GBZ from the characteristic equation   0 = det σz HBdG (β) − E    1 2 1 2  − t 2 β 2 + β −2 − μ2 + E 2 +  − t 2 cos 2φ = 4 2 − (μt cos φ − it E sin φ) β − (μt cos φ + it E sin φ) β −1

(6.24)

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6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

by applying the condition (6.16). The resulting GBZ is not a unit circle in the complex β plane. This is why it exhibits the non-Hermitian skin effect. When μ = 0, it is shown that σz HBdG (β) can be transformed into the block diagonal matrix form as P (σz HBdG (β)) P

−1

 =

H+ (β) O O H− (β)

 (6.25)

by the similarity transformation given by 1 P=√ 2C (C + t cos φ)



C + t cos φ −i i C + t cos φ

 , (6.26)

where C =

 t 2 cos2 φ − 2 . Then, the diagonal elements of Eq. (6.25) are given by

H± (β) =

  1   t sin φ β − β −1 ± t 2 cos2 φ − 2 β + β −1 . 2i 2

(6.27)

In other words, Eq. (6.24) can be factorized into two irreducible algebraic equations for β and E. Therefore, in this case, this two-band model is decoupled into two systems described by H± (β), which are the simple models given in Chap. 3. We note that the condition for each band. Namely, they are   for the GBZ is also split ± are the solutions of the equations given by β1±  = β2±  for each band, where β1,2 H± (β) − E = 0. Before going to the discussion of non-Hermitian properties in this system, let us introduce the energy winding number [9, 10], which tells us the topological origin of the non-Hermitian skin effect. Under a periodic boundary condition, the topological invariant can be defined as  2π d 1 log det [H (k) − E] , (6.28) dk W (E) = 2πi 0 dk where k is the conventional Bloch wave number,  and H (k) is the conventional Bloch Hamiltonian written as H (k) ≡ H β = eik , k ∈ R from our non-Bloch matrix H (β). For a given value of the reference energy E ∈ C, when the energy winding number (6.28) takes nonzero values, the non-Hermitian skin effect occurs as manifestation of the non-Hermitian topology. On the other hand, when W (E) = 0 for any reference energies, the non-Hermitian skin effect does not occur. Thus, by using Eq. (6.28), we can investigate the behavior of a non-Hermitian system. In the following, we discuss the physics of the model for three different cases of the values of the parameter φ.

6.2 Bosonic Kitaev-Majorana Chain

79

Case I: φ = 0

6.2.1.1

In the case of μ = 0, the energy winding number (6.28) for each √ band is always zero because the energy eigenvalues take either of the real values ± t 2 − 2 cos k for k ∈ R. Furthermore, the energy winding number takes zero even in the case of μ = 0. Therefore, the non-Hermitian skin effect does not occur, and the resulting GBZ is always a unit circle, identical with Hermitian systems. Case II: φ = π/2

6.2.1.2

First of all, we begin with the case of μ = 0. Then, from Eq. (6.27), the energy winding number (6.28) for each band with H± (β) is ±1 for any values of the reference energy within the region surrounded by the ellipse E = t sin k + i cos k, k ∈ R on the complex energy plane, which is the energy eigenvalue for a periodic chain, shown in black in Fig. 6.1a. Therefore, the non-Hermitian skin effect occurs in an open chain. In fact, we can confirm that the energy spectrum in an open chain completely differs from that in a periodic chain as shown in Fig. 6.1a, and the GBZ for each band is not a unit circle but a circle with the radius

Im(E)

0.5

(a) µ=0

Periodic Open

0

Open Periodic

(c)

-0.5 -1

1

0

Re(E)

µ=10-15 µ=10-8 µ=10-3

(b)

Im(β)

0.5

Im(E)

µ≠0

1

0

-0.5

µ=0

-1

-1

-1

0

1

Re(E)

1 Re(β)

Fig. 6.1 Eigenenergies and generalized Brillouin zone (GBZ) in the bosonic Kitaev-Majorana chain with various values of the parameter μ in the case of φ = π/2. We set the parameters to be t = 1 and  = 0.7. a Energy spectra in an open chain (red) and in a periodic chain (black) with μ = 0. b GBZ in the case of μ = 0 (green andblue) and circle   green  (blue)  of μ = 0 (black). The describes the GBZ for the band with H+ (β) H− (β) and has the radius β +  (β − ). c Energy levels in a finite periodic chain and in a finite open chain with μ = 10−15 , 10−8 , and 10−3 . We set the system size to be L = 50. In this case, the critical value can be obtained as μ0 10−12 . We note that the energy levels for a periodic chain with μ = 10−15 , 10−8 , and 10−3 almost overlap in the figure

80

6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

 ± β  =

t ± t ∓

(6.29)

for the bands with H± (β) as shown in Fig. 6.1b. This means that the eigenstates from H+ (β) and those from H− (β) are localized at the opposite ends of an open chain. While the non-Hermitian skin effect appears when μ = 0, it becomes unstable against perturbations by the infinitesimal values of the parameter μ [11, 12]. When μ becomes nonzero, the matrix σz HBdG (β) cannot be written as the block-diagonal matrix form (6.25), which means that the two localized eigenstates couple with each other. Namely, this perturbation couples the two bands with the energy winding number being equal to ±1, leading to a trivial value of the total energy winding number, and thus the non-Hermitian skin effect is immediately suppressed. As a result, the GBZ becomes a unit circle when μ = 0 as shown in Fig. 6.1b. In this case, the Bloch wave number takes real values, and the energy spectrum in an open chain coincides with that in a periodic chain. The transition from the nontrivial phase to the trivial phase discontinuously occurs as μ becomes nonzero. On the other hand, in a finite open chain, the energy levels continuously approach those in a finite periodic chain as the value of μ continuously increases, with a rapid change around the critical value μ0 , as shown in Fig. 6.1c. Here, from Eq. (6.29), the order of μ0 is expected to be [11]    L (6.30) μ0 /t O β −  , where L is a system size. For example, when t = 1,  = 0.7, and L = 50 adopted in Fig. 6.1c, we get μ0 10−12 , and indeed, one can confirm this value of μ0 in a finite open chain; the energy levels with μ = 10−15 < μ0 largely deviate from those with μ = 10−8 and 10−3 exceeding μ0 . Thus, the non-Hermitian skin effect exhibits instability against the infinitesimal values of the perturbation.

6.2.1.3

Case III: φ = 0, π/2

When μ = 0, whether the non-Hermitian skin effect occurs or not depends on the values of the system parameters. When t |cos φ| > , the non-Hermitian skin effect disappears because the energy winding number (6.28) becomes zero for any reference energies.  This is because the energy eigenvalues for k ∈ R take the real values t sin φ sin k ± t 2 cos2 φ − 2 cos k. On the other hand, when t |cos φ| < , the system exhibits the non-Hermitian skin effect. In fact, the above discussion is consistent with the numerical results as shown in Fig. 6.2a1,b1,a4 and b4. Next, we focus on the case of μ = 0. When μ is a positive infinitesimal, μ = +0, the GBZ is a unit circle, regardless of the value of the parameter . It is seen in Fig. 6.2b2 and b5 for μ = 0.01. Here, we note that because μ is small but finite, the GBZ slightly deviates from a unit circle, but this deviation is tiny when μ  O (t, ). Accordingly, the energy spectra in an open chain are almost identical with those in

6.2 Bosonic Kitaev-Majorana Chain

81

Fig. 6.2 Energy spectra and generalized Brillouin zone (GBZ) in the bosonic Kitaev-Majorana chain with various values of the parameter μ in the case of φ = π/3. We set the parameters to be t = 1. a1–a6 Energy spectra in an open chain (red) and in a periodic chain (black). In (a3) and (a6), the region in green (blue) represents that the energy winding number W (E) takes +1 (−1). b1–b6 GBZ with the colored lines. The black broken line expresses a unit circle, meaning the conventional Brillouin zone. c1–c3 Schematic figures of the reentrant behavior with the energy winding number through an increase of the value of the parameter μ. At c1 μ = 0, the two bands with W (E) = ±1 are decoupled. When we increase μ, they couple with each other, leading to W (E) = 0 as shown in (c2). A further increase of μ leads to the regions with the nonzero values of the energy winding number because of the deviation between the two energy bands in a periodic chain as shown in (c3). The region in green (blue) describes the region with the energy winding number being 1 (−1)

82

6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

a periodic chain, shown in Fig. 6.2a2 and a5, and the non-Hermitian skin effect is almost absent. As the value of μ increases, the deviation of the GBZ from a unit circle becomes prominent [Fig. 6.2b3 and b6]. Therefore, in particular, when t |cos φ| < , as the value of μ increases, the non-Hermitian skin effect disappears once and it reoccurs as shown in Fig. 6.2a4–a6 and b4–b6. In this sense, we call this phenomenon reentrant behavior. It is noted that the reappearance of the nonHermitian skin effect occurs in μ > 0 as crossover, and not phase transition. The GBZ in Fig. 6.2b3 and b6 partially overlaps a unit circle shown in a black dashed line. This means that the Bloch wave number takes real values because the non-Bloch BdG matrix (6.23) is positive definite in some regions on the complex β plane. On the other hand, the positive definiteness of the non-Bloch BdG matrix is partially broken in the other regions, and the shape of the GBZ deviates from a unit circle in such regions. Thus, the Hermiticity and the non-Hermiticity can coexist in a set of system parameters. Now, we explain the reason for the reentrant behavior of the non-Hermitian skin effect upon changing the value of μ as shown in Fig. 6.2c1–c3. As mentioned above, when μ = 0, this system can be regarded as two decoupled simple models [Fig. 6.2c1]. In this case, for any values of the reference energy E in the region surrounded by the energy band, the values of the energy winding number W (E) are equal to ±1 for the two decoupled bands, leading to one skin mode each. On the other hand, for the infinitesimal values of μ, the two bands couple with each other, and the total energy winding number in this system is merely summed over the two bands to approximately become zero [Fig. 6.2c2]. Therefore, the non-Hermitian skin effect almost disappears, as we have already seen in Fig. 6.2a5 and b5. This disappearance gradually occurs when the system size is not large, and it becomes sharp at μ being infinitesimal in the limit of a large system size. Furthermore, as the value of μ increases, the splitting of the two bands becomes significant, and the region with W (E) = 0 appears [Fig. 6.2c3]. Therefore, these regions give rise to the two skin modes localized at both ends of an open chain. We note that in the case of φ = π/2, this deviation does not occur because the two bands are accidentally degenerate for any values of μ. Finally, in order to see how the non-Hermitian skin effect appears in real space, we show the numerical result of the absolute value of the real-space distribution of the coefficients u κj and v κj included in the Bogoliubov transformation (6.19) with t = 1,  = 0.7, φ = π/3, and μ = 0.01 adopted in Fig. 6.2a5 and b5, and μ = 0.1 adopted in Fig. 6.2a6 and b6. When μ = 0.01, the non-Hermitian skin effect almost does not occur, and their distribution is almost uniform [Fig. 6.3a]. In contrast, when μ = 0.1, u κj and v κj at the boundaries have larger values than any other sites [Fig. 6.3b]. This indicates that the non-Hermitian skin effect reoccurs as the values of μ increase from μ = 0.01. Therefore, this numerical result is consistent with the above discussion.

6.2 Bosonic Kitaev-Majorana Chain

83

Fig. 6.3 Real-space distribution of the absolute values of the coefficients u κj and v κj for κ = 1, . . . , 50 in the Bogoliubov transformation with t = 1,  = 0.7, φ = π/3, and (a) μ = 0.01 and (b) μ = 0.1, respectively. We set the system size to be L = 50.

6.2.2 Analytical Representation of the Bogoliubov Transformation In this subsection, we study the Bogoliubov transformation in a special case, i.e., μ = 0 and φ = π/2. We assume t >  in the following. In this case, from Eq. (6.27), we can get the analytical form of the Bogoliubov transformation in a finite open chain with the system size L. The values of βm± (m = 1, 2) are explicitly written as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ β+ = i 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ β1− = i

t +  iθ e , β2+ = i t −

t +  −iθ e , t −

t −  iθ e , β2− = i t +

t −  −iθ e , t +

(6.31)

√ where θ ∈ R. We note that the system has the energy spectrum as E = t 2 − 2 cos k. Then, we can determine the coefficients included in Eq. (6.19) so as to satisfy the conditions (6.20) and (6.21) and can get the analytical representation of the quasiparticle-creation operator as αn† =

L

4  

u κ,m (βm ) j a †j − v κ,m (βm ) j a j j=1 m=1

=

L !

 + j  + j  − j  − j  1 β1 − β2 + β1 − β2 aj 2i 2 (L + 1) j=1     j   j   j  †" j aj − β1+ − β2+ − β1− + β2−

√

 2 j i sin (θn j) cosh (r j) a †j − sin (θn j) sinh (r j) a j , (6.32) L + 1 j=1 L

=

84

6 Non-Bloch Band Theory in Bosonic Bogoliubov-de Gennes Systems

where e2r =

t + , t −

(6.33)

and θn = nπ/ (L + 1) (n = 1, . . . , L). We note that these results obtained here agree with those in Ref. [8]. Importantly, we can systematically get the Bogoliubov transformation in terms of the non-Bloch band theory.

6.3 Summary In this chapter, we study the non-Bloch band theory in a bosonic BdG system. Although the original Hamiltonian is Hermitian, the system is intrinsically nonHermitian, and it shows various features unique to non-Hermitian systems, such as the non-Hermitian skin effect. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the non-Hermitian skin effect. Interestingly, this model with φ = 0, π/2 exhibits the reentrant behavior of the non-Hermitian skin effect, which has not been found. Furthermore, the system has the coexistence of Hermiticity and non-Hermiticity. We emphasize that the feature can be accessible only via the non-Bloch band theory. Finally, we note that when the value of the parameter μ is sufficiently large, the system behaves as a Hermitian system because the non-Bloch BdG matrix (6.23) becomes positive definite.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Kawabata K, Shiozaki K, Ueda M, Sato M (2019) Phys. Rev. X 9:041015 Colpa J (1978) Physica A 93:327 Shindou R, Ohe J-I, Matsumoto R, Murakami S, Saitoh E (2013) Phys. Rev. B 87:174402 Shindou R, Matsumoto R, Murakami S, Ohe J-I (2013) Phys. Rev. B 87:174427 Matsumoto R, Shindou R, Murakami S (2014) Phys. Rev. B 89:054420 Lieu S (2018) Phys. Rev. B 98:115135 Brody DC (2104) J. Phys. A 47:035305 McDonald A, Pereg-Barnea T, Clerk AA (2018) Phys. Rev. X 8:041031 Okuma N, Kawabata K, Shiozaki K, Sato M (2020) Phys. Rev. Lett. 124:086801 Zhang K, Yang Z, Fang C (2020) Phys. Rev. Lett. 125:126402 Okuma N, Sato M (2019) Phys. Rev. Lett. 123:097701 Li L, Lee CH, Mu S, Gong J (2020) Nat. Commun. 11:5491

Chapter 7

Summary and Outlook

In non-Hermitian systems, the behavior of the eigenstates under an open boundary condition is different from that under a periodic boundary condition. This phenomenon occurs as a manifestation of the non-Hermitian skin effect, which is unique to non-Hermitian systems. In Chap. 3, we showed that the non-Hermitian skin effect causes the difference between the energy eigenvalues under an open boundary condition and those under a periodic boundary condition and the localization of the bulk eigenstates in the simple model. The complex eigenenergies in a periodic chain drastically differ from the real eigenvalues in an open chain. Furthermore, while the eigenstates in a periodic chain extend over the whole system, the eigenstates in an open chain are exponentially localized at the boundary. These results indicate that we should take into account the non-Hermitian skin effect in order to study the general properties of non-Hermitian systems, such as the bulk-edge correspondence. Because of the non-Hermitian skin effect, the Bloch wave number k takes complex values to describe eigenstates in a long open chain. Then the value of β = eik is confined to closed loops on the complex plane so as to reproduce energy spectra. This loop is a generalization of the conventional Brillouin zone and is called the generalized Brillouin zone (GBZ). Hence, in Chap. 4, we established the non-Bloch band theory in order to determine the GBZ for the complex Bloch wave number in a one-dimensional (1D) tight-binding non-Hermitian system. In this theory, we showed the condition for the GBZ by using the  non-Bloch matrix H (β). Namely, for the characteristic equation det H (β) − E = 0, which is an algebraic equation for β with an even degree 2M, the condition for the GBZ can be written as |β M | = |β M+1 | when the characteristic equation has 2M solutions β1 , . . . , β2M with |β1 | ≤ · · · ≤ |β2M |. Then, the trajectories of β M and β M+1 are the GBZ. Throughout the discussion of this chapter, we revealed some properties of the GBZ. Firstly, in a multi-band non-Hermitian system, each of the energy bands has its own GBZ. In fact, we demonstrate the splitting of the GBZ in the non-Hermitian ladder model II. Secondly, the absolute values of β on the GBZ can take both values more than 1 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2_7

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and values less than 1. Thirdly, the GBZ can have the cusps. Finally, we emphasize that our main conclusion is physically reasonable in several aspects. Namely, the condition for the GBZ is independent of any boundary conditions in an open chain. Furthermore, in the Hermitian limit, this condition reduces to a well-known result, i.e., |β M | = |β M+1 | = 1, meaning that the Bloch wave number becomes real, and we can get the conventional Brillouin zone. In terms of the non-Bloch band theory, we can also establish the bulk-edge correspondence in non-Hermitian systems. In the non-Hermitian Su-Schrieffer-Heeger (SSH) model, we showed the bulk-edge correspondence between the topological invariant defined by the GBZ and the existence of the topological edge states. Here, we define the Z topological invariant, called the winding number, similar to the method of the Q matrix in Hermitian systems explained in Chap. 2. While this method can be straightforwardly extended to a two-band non-Hermitian system, we commented that in a multi-band non-Hermitian system, it is difficult to define the winding number in general. In Chap. 4, we found that the gapless phases extend over the phase diagram in the non-Hermitian SSH model, and in this phase, the system has the exceptional points. Hence, in Chap. 5, we investigated the mechanism for the appearance of this gapless phase. As a result, we showed that in a 1D non-Hermitian system with both the sublattice symmetry (SLS) and the time-reversal symmetry (TRS), such as the non-Hermitian SSH model, the topological semimetal (TSM) phase with exceptional points is stabilized because of the unique features of the GBZ. Namely, as the values of system parameters change, the GBZ is deformed so that the exceptional points always lie on the GBZ, and the systems remain gapless. Then, the TSM phase appears as an intermediate phase between the normal insulator phase and the topological insulator phase, where insulating phases are topologically classified in terms of the winding number. This can be proved by relating the change of the value of the winding number with the creation and annihilation of the exceptional points. Furthermore, we also found that each energy band is divided into three regions, depending on the symmetry of the eigenstates. In fact, in the non-Hermitian SSH model, we demonstrated that the energy band is divided into the TRS-unbroken region, the STS-unbroken region, and the TRS/STS-broken region. These three regions are separated from each other by the cusps and the exceptional points in the GBZ. In Chap. 6, we studied the non-Bloch band theory in a bosonic Bogoliubov-de Gennes (BdG) system in order to study non-Hermiticity of this system. We showed that from the non-Bloch BdG matrix, one can get the GBZ and the Bogoliubov transformation. In terms of the GBZ, we investigated the bosonic Kitaev-Majorana chain. In a special case, we can divide this two-band system into two single-band systems, which are independent of each other. In this case, the two systems have skin modes localized at the ends opposite to each other. Then, by adding the perturbation which couples the two skin modes, the non-Hermitian skin effect disappears. The disappearance of the non-Hermitian skin effect is caused by any infinitesimal values of the perturbation, and therefore, the whole system exhibits the infinitesimal instability of the non-Hermitian skin effect. Nevertheless, as the strength of the perturbation increases, the system shows the reentrant behavior of the non-Hermitian skin effect.

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Thus, we found that the non-Bloch band theory can reveal rich aspects of the nonHermitian skin effect. Finally, we stress that although the original Hamiltonian is Hermitian, the bosonic BdG system is intrinsically non-Hermitian. This is because the original Hamiltonian is locally a Hermitian operator but not a self-adjoint operator due to existence of boundaries. In the following, we mention theoretical researches on the non-Hermitian skin effect, experimental realization of it, and extension of the non-Bloch band theory. After the proposal of the non-Hermitian skin effect, its various aspects have been intensively studied in both theories and experiments. In theories, the topological structure of the non-Hermitian skin effect and its applications have been revealed in many previous works [1–15]. In experiments, the non-Hermitian skin effect was observed in various systems. Reference [16] realized a nonreciprocal tight-binding model in a classical spring-mass system similar to the simple model given in Chap. 3, and it observed spatially asymmetric standing waves. After that, in Ref. [17], the non-Hermitian SSH model with short-range hopping amplitude was realized, and the non-Hermitian skin effect was demonstrated by investigating the nonunitary quantum walk dynamics. Furthermore, Ref. [18] also realized the non-Hermitian SSH model by using an electric circuit. The previous work showed the difference between the energy eigenvalues in an open chain and those in a periodic chain through observation of the complex admittance. Thus, an electric circuit is useful for realizing non-Hermitian systems and for investigating novel phenomena caused by the nonHermitian skin effect. Indeed, by making the cylinder geometry in a two-dimensional electric circuit system, Ref. [19] experimentally showed that the localization position of the eigenstates changes, depending on the wave number along the direction with a periodic boundary condition. Besides, in a cold-atom system, the nonreciprocal transport was experimentally observed by using the dissipative Aharonov-Bohm (AB) ring [20]. The previous work further theoretically proposed that it is possible to realize the non-Hermitian SSH model by connecting some dissipative AB rings. Thus, so far, most of the research works have mainly focused on studies based on non-Hermitian models and have investigated the non-Hermitian properties by using various artificial physical systems. On the other hand, the investigation of nonHermiticity is more complicated in other physical systems. For example, in a strongly correlated electron system, the localization of bulk eigenstates is forbidden because of the Pauli exclusion principle [8]. This proposal is consistent with the result obtained in terms of the Green function [21]. Namely, these previous works showed that in a strongly correlated electron system, the existence of boundaries almost does not affect the behavior of eigenstates in the bulk. Nevertheless, Ref. [21] also proposed that measurement by the angle-resolved photoemission spectroscopy depends on boundaries in principle because the spectral function exhibits the sensitivity of the boundaries. This indicates that one can observe some phenomena caused by the non-Hermitian skin effect in an electron system. It is left as a future work. Here, we comment on the experimental realization of the TSM phase as discussed in Chap. 5. Reference [15] theoretically proposed the non-Hermitian SSH model with long-range hopping amplitude that can be realized in an electric circuit. Therefore, we expect that our theory can be verified. Since the only restrictions are both the SLS

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and the TRS, there remains much room for the choice of parameter values, toward experimental verification. The application of the non-Bloch band theory have been theoretically proposed in some previous works [12, 22–28]. We further comment on application of the nonBloch band theory in bosonic BdG systems. So far, non-Hermitian physics, such as dynamical instability in some cold-atom systems [29–33], which are described by a BdG Hamiltonian, has been intensively investigated. In Chap. 6, we showed that non-Hermitian phenomena in such systems are accessible via the non-Bloch band theory. Therefore, we expect that the non-Bloch band theory paves a way for the implementation of various non-Hermitian phenomena, such as the non-Hermitian skin effect, in many bosonic BdG systems, not only a cold-atom system but also a magnon system, a phonon system, and a photon system.

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Curriculum Vitae

Kazuki Yokomizo Condensed Matter Theory Laboratory RIKEN 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan e-mail: [email protected]

Appointments April 2021–present January 2021–March 2021 April 2018–December 2020

JSPS research fellow (PD), RIKEN JSPS research fellow (PD), Tokyo Institute of Technology JSPS research fellow (DC1), Tokyo Institute of Technology

Education 2020 2018 2016

Ph.D in Science, Tokyo Institute of Technology M.E. in Science, Tokyo Institute of Technology B.E. in Science, Tokyo Institute of Technology

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Yokomizo, Non-Bloch Band Theory of Non-Hermitian Systems, Springer Theses, https://doi.org/10.1007/978-981-19-1858-2

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Refereed Papers 1. Kazuki Yokomizo and Shuichi Murakami, “Scaling rule in critical non-Hermitian skin effect”, Phys. Rev. B 104, 165117 (2021). 2. Ryo Okugawa, Ryo Takahashi, and Kazuki Yokomizo, “Non-Hermitian band topology with generalized inversion symmetry”, Phys. Rev. B 103, 205205 (2021). 3. Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch band theory in bosonic Bogoliubov-de Gennes systems”, Phys. Rev. B 103, 165123 (2021). 4. Ryo Okugawa, Ryo Takahshi, and Kazuki Yokomizo, “Second-order non-Hermitian skin effects”, Phys. Rev. B 102, 241202(R) (2020). 5. Kazuki Yokomizo and Shuichi Murakami, “Topological semimetal phase with exceptional points in one-dimensional nonHermitian systems”, Phys. Rev. Research 2, 043045 (2020). 6. Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch band theory and bulk-edge correspondence in non-Hermitian systems, Prog. Theor. Exp. Phys. 2020, 12A102 (2020). 7. Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch Band Theory of Non-Hermitian Systems”, Phys. Rev. Lett. 123, 066404 (2019). 8. Kazuki Yokomizo, Hiroaki Yamada, and Shuichi Murakami, “Noda-line semimetal superlattices”, J. Phys.: Condens. Matter 30, 505301 (2018). 9. Kazuki Yokomizo and Shuichi Murakami, Topological phases in a Weyl semimetal multilayer, Phys. Rev. B 95, 155101 (2017).