Homotopy characterization of non-Hermitian Hamiltonians
Abstract
We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called “point-gap” and “line-gap” schemes. However, simple Hamiltonians without band degeneracies can be constructed which correspond to neither of the two schemes. Here, we resolve this shortcoming of the existing classifications by developing the most general topological characterization of non-Hermitian bands for systems without a symmetry. Our approach, which is based on homotopy theory, makes no particular assumptions on the band gap, and predicts significant extensions to the previous classification frameworks. In particular, we show that the one-dimensional invariant generalizes from $$\mathbb{Z}$$ winding number to the non-Abelian braid group, and that depending on the braid group invariants, the two-dimensional invariants can be cyclic groups $$\mathbb{Z}_n$$ (rather than $$\mathbb{Z}$$ Chern number). Finally, we interpret these results in terms of a correspondence with gapless systems, and we illustrate them in terms of analogies with other problems in band topology, namely, the fragile topological invariants in Hermitian systems and the topological defects and textures of nematic liquids.
- Authors:
-
[1];
- Stanford Univ., CA (United States)
- Stanford Univ., CA (United States); Paul Scherrer Inst. (PSI), Villigen (Switzerland); Univ. of Zurich (Switzerland)
- Publication Date:
- Research Org.:
- Univ. of California, Oakland, CA (United States)
- Sponsoring Org.:
- USDOD; National Science Foundation (NSF); USDOE Office of Science (SC), High Energy Physics (HEP); Simons Foundation; Gordon and Betty Moore Foundation; Swiss National Science Foundation (SNF)
- OSTI Identifier:
- 1803752
- Grant/Contract Number:
- SC0019380; N00014-17-1-3030; CBET-1641069; GBMF4302; 185806
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review B
- Additional Journal Information:
- Journal Volume: 101; Journal Issue: 20; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Materials Science; Physics
Citation Formats
Wojcik, Charles C., Sun, Xiao-Qi, Bzdušek, Tomáš, and Fan, Shanhui. Homotopy characterization of non-Hermitian Hamiltonians. United States: N. p., 2020.
Web. doi:10.1103/physrevb.101.205417.
Wojcik, Charles C., Sun, Xiao-Qi, Bzdušek, Tomáš, & Fan, Shanhui. Homotopy characterization of non-Hermitian Hamiltonians. United States. https://doi.org/10.1103/physrevb.101.205417
Wojcik, Charles C., Sun, Xiao-Qi, Bzdušek, Tomáš, and Fan, Shanhui. Fri .
"Homotopy characterization of non-Hermitian Hamiltonians". United States. https://doi.org/10.1103/physrevb.101.205417. https://www.osti.gov/servlets/purl/1803752.
@article{osti_1803752,
title = {Homotopy characterization of non-Hermitian Hamiltonians},
author = {Wojcik, Charles C. and Sun, Xiao-Qi and Bzdušek, Tomáš and Fan, Shanhui},
abstractNote = {We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called “point-gap” and “line-gap” schemes. However, simple Hamiltonians without band degeneracies can be constructed which correspond to neither of the two schemes. Here, we resolve this shortcoming of the existing classifications by developing the most general topological characterization of non-Hermitian bands for systems without a symmetry. Our approach, which is based on homotopy theory, makes no particular assumptions on the band gap, and predicts significant extensions to the previous classification frameworks. In particular, we show that the one-dimensional invariant generalizes from $\mathbb{Z}$ winding number to the non-Abelian braid group, and that depending on the braid group invariants, the two-dimensional invariants can be cyclic groups $\mathbb{Z}_n$ (rather than $\mathbb{Z}$ Chern number). Finally, we interpret these results in terms of a correspondence with gapless systems, and we illustrate them in terms of analogies with other problems in band topology, namely, the fragile topological invariants in Hermitian systems and the topological defects and textures of nematic liquids.},
doi = {10.1103/physrevb.101.205417},
journal = {Physical Review B},
number = 20,
volume = 101,
place = {United States},
year = {Fri May 15 00:00:00 EDT 2020},
month = {Fri May 15 00:00:00 EDT 2020}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Bulk-boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry
journal, February 2019
- Jin, L.; Song, Z.
- Physical Review B, Vol. 99, Issue 8
Topological insulators and superconductors
journal, October 2011
- Qi, Xiao-Liang; Zhang, Shou-Cheng
- Reviews of Modern Physics, Vol. 83, Issue 4
The topological theory of defects in ordered media
journal, July 1979
- Mermin, N. D.
- Reviews of Modern Physics, Vol. 51, Issue 3
Hopf insulators and their topologically protected surface states
journal, November 2013
- Deng, D. -L.; Wang, S. -T.; Shen, C.
- Physical Review B, Vol. 88, Issue 20
Band Topology and Linking Structure of Nodal Line Semimetals with Monopole Charges
journal, September 2018
- Ahn, Junyeong; Kim, Dongwook; Kim, Youngkuk
- Physical Review Letters, Vol. 121, Issue 10
-Symmetric Real Dirac Fermions and Semimetals
journal, February 2017
- Zhao, Y. X.; Lu, Y.
- Physical Review Letters, Vol. 118, Issue 5
Experimental realization of a Weyl exceptional ring
journal, June 2019
- Cerjan, Alexander; Huang, Sheng; Wang, Mohan
- Nature Photonics, Vol. 13, Issue 9
Quantum Oscillation from In-Gap States and a Non-Hermitian Landau Level Problem
journal, July 2018
- Shen, Huitao; Fu, Liang
- Physical Review Letters, Vol. 121, Issue 2
Unconventional superconductivity in magic-angle graphene superlattices
journal, March 2018
- Cao, Yuan; Fatemi, Valla; Fang, Shiang
- Nature, Vol. 556, Issue 7699
Optimal Topological Test for Degeneracies of Real Hamiltonians
journal, February 2004
- Johansson, Niklas; Sjöqvist, Erik
- Physical Review Letters, Vol. 92, Issue 6
Non-Hermitian dynamics of slowly varying Hamiltonians
journal, July 2018
- Wang, Hailong; Lang, Li-Jun; Chong, Y. D.
- Physical Review A, Vol. 98, Issue 1
Observation of bulk Fermi arc and polarization half charge from paired exceptional points
journal, January 2018
- Zhou, Hengyun; Peng, Chao; Yoon, Yoseob
- Science, Vol. 359, Issue 6379
Stability of Fermi Surfaces and Theory
journal, June 2005
- Hořava, Petr
- Physical Review Letters, Vol. 95, Issue 1
Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks
journal, September 2017
- Zhan, Xiang; Xiao, Lei; Bian, Zhihao
- Physical Review Letters, Vol. 119, Issue 13
Topological Phases of Non-Hermitian Systems
journal, September 2018
- Gong, Zongping; Ashida, Yuto; Kawabata, Kohei
- Physical Review X, Vol. 8, Issue 3
Stiefel–Whitney classes and topological phases in band theory
journal, October 2019
- Ahn, Junyeong; Park, Sungjoon; Kim, Dongwook
- Chinese Physics B, Vol. 28, Issue 11
Observation of topological edge states in parity–time-symmetric quantum walks
journal, July 2017
- Xiao, L.; Zhan, X.; Bian, Z. H.
- Nature Physics, Vol. 13, Issue 11
Characteristic Classes. (AM-76)
book, December 1974
- Milnor, John; Stasheff, James D.
- Princeton University Press
Experimental observation of Weyl points
journal, July 2015
- Lu, L.; Wang, Z.; Ye, D.
- Science, Vol. 349, Issue 6248
Selective enhancement of topologically induced interface states in a dielectric resonator chain
journal, April 2015
- Poli, Charles; Bellec, Matthieu; Kuhl, Ulrich
- Nature Communications, Vol. 6, Issue 1
Synthetic dimension in photonics
journal, January 2018
- Yuan, Luqi; Lin, Qian; Xiao, Meng
- Optica, Vol. 5, Issue 11
Flat band in disorder-driven non-Hermitian Weyl semimetals
journal, January 2018
- Zyuzin, A. A.; Zyuzin, A. Yu.
- Physical Review B, Vol. 97, Issue 4
Experimental observation of optical Weyl points and Fermi arc-like surface states
journal, March 2017
- Noh, Jiho; Huang, Sheng; Leykam, Daniel
- Nature Physics, Vol. 13, Issue 6
Topologically protected bound states in photonic parity–time-symmetric crystals
journal, December 2016
- Weimann, S.; Kremer, M.; Plotnik, Y.
- Nature Materials, Vol. 16, Issue 4
Colloquium : Disclination loops, point defects, and all that in nematic liquid crystals
journal, April 2012
- Alexander, Gareth P.; Chen, Bryan Gin-ge; Matsumoto, Elisabetta A.
- Reviews of Modern Physics, Vol. 84, Issue 2
Fragile Topology and Wannier Obstructions
journal, September 2018
- Po, Hoi Chun; Watanabe, Haruki; Vishwanath, Ashvin
- Physical Review Letters, Vol. 121, Issue 12
Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle
journal, April 2019
- Ahn, Junyeong; Park, Sungjoon; Yang, Bohm-Jung
- Physical Review X, Vol. 9, Issue 2
Localization Transitions in Non-Hermitian Quantum Mechanics
journal, July 1996
- Hatano, Naomichi; Nelson, David R.
- Physical Review Letters, Vol. 77, Issue 3
Non-Abelian band topology in noninteracting metals
journal, August 2019
- Wu, QuanSheng; Soluyanov, Alexey A.; Bzdušek, Tomáš
- Science, Vol. 365, Issue 6459
Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
journal, January 2017
- Leykam, Daniel; Bliokh, Konstantin Y.; Huang, Chunli
- Physical Review Letters, Vol. 118, Issue 4
Edge States and Topological Invariants of Non-Hermitian Systems
journal, August 2018
- Yao, Shunyu; Wang, Zhong
- Physical Review Letters, Vol. 121, Issue 8
Non-Hermitian Topological Invariants in Real Space
journal, December 2019
- Song, Fei; Yao, Shunyu; Wang, Zhong
- Physical Review Letters, Vol. 123, Issue 24
Topological phases and edge states in a non-Hermitian trimerized optical lattice
journal, September 2017
- Jin, L.
- Physical Review A, Vol. 96, Issue 3
Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states
journal, October 2016
- Chen, Wen-Jie; Xiao, Meng; Chan, C. T.
- Nature Communications, Vol. 7, Issue 1
Topological nodal line semimetals
journal, November 2016
- Fang, Chen; Weng, Hongming; Dai, Xi
- Chinese Physics B, Vol. 25, Issue 11
Topological insulators and superconductors: tenfold way and dimensional hierarchy
journal, June 2010
- Ryu, Shinsei; Schnyder, Andreas P.; Furusaki, Akira
- New Journal of Physics, Vol. 12, Issue 6
Classification of Exceptional Points and Non-Hermitian Topological Semimetals
journal, August 2019
- Kawabata, Kohei; Bessho, Takumi; Sato, Masatoshi
- Physical Review Letters, Vol. 123, Issue 6
Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry
journal, November 2019
- Bouhon, Adrien; Black-Schaffer, Annica M.; Slager, Robert-Jan
- Physical Review B, Vol. 100, Issue 19
Anatomy of skin modes and topology in non-Hermitian systems
journal, May 2019
- Lee, Ching Hua; Thomale, Ronny
- Physical Review B, Vol. 99, Issue 20
Quantized Hall Conductance in a Two-Dimensional Periodic Potential
journal, August 1982
- Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.
- Physical Review Letters, Vol. 49, Issue 6
Symmetry and Topology in Non-Hermitian Physics
journal, October 2019
- Kawabata, Kohei; Shiozaki, Ken; Ueda, Masahito
- Physical Review X, Vol. 9, Issue 4
Topological Order and the Quantum Spin Hall Effect
journal, September 2005
- Kane, C. L.; Mele, E. J.
- Physical Review Letters, Vol. 95, Issue 14, Article No. 146802
Non-Hermitian Boundary Modes and Topology
journal, February 2020
- Borgnia, Dan S.; Kruchkov, Alex Jura; Slager, Robert-Jan
- Physical Review Letters, Vol. 124, Issue 5
Colloquium: Topological insulators
journal, November 2010
- Hasan, M. Z.; Kane, C. L.
- Reviews of Modern Physics, Vol. 82, Issue 4, p. 3045-3067
Non-Hermitian Chern Bands
journal, September 2018
- Yao, Shunyu; Song, Fei; Wang, Zhong
- Physical Review Letters, Vol. 121, Issue 13
Faithful tight-binding models and fragile topology of magic-angle bilayer graphene
journal, May 2019
- Po, Hoi Chun; Zou, Liujun; Senthil, T.
- Physical Review B, Vol. 99, Issue 19
Topological Band Theory for Non-Hermitian Hamiltonians
journal, April 2018
- Shen, Huitao; Zhen, Bo; Fu, Liang
- Physical Review Letters, Vol. 120, Issue 14
Observation of a Topological Transition in the Bulk of a Non-Hermitian System
journal, July 2015
- Zeuner, Julia M.; Rechtsman, Mikael C.; Plotnik, Yonatan
- Physical Review Letters, Vol. 115, Issue 4
Topological Surface States in Three-Dimensional Magnetic Insulators
journal, October 2008
- Moore, Joel E.; Ran, Ying; Wen, Xiao-Gang
- Physical Review Letters, Vol. 101, Issue 18
Moire bands in twisted double-layer graphene
journal, July 2011
- Bistritzer, R.; MacDonald, A. H.
- Proceedings of the National Academy of Sciences, Vol. 108, Issue 30
Correlated insulator behaviour at half-filling in magic-angle graphene superlattices
journal, March 2018
- Cao, Yuan; Fatemi, Valla; Demir, Ahmet
- Nature, Vol. 556, Issue 7699
Quasieigenstate Coalescence in an Atom-Cavity Quantum Composite
journal, April 2010
- Choi, Youngwoon; Kang, Sungsam; Lim, Sooin
- Physical Review Letters, Vol. 104, Issue 15
Spawning rings of exceptional points out of Dirac cones
journal, September 2015
- Zhen, Bo; Hsu, Chia Wei; Igarashi, Yuichi
- Nature, Vol. 525, Issue 7569
Topological Classification of Crystalline Insulators through Band Structure Combinatorics
journal, December 2017
- Kruthoff, Jorrit; de Boer, Jan; van Wezel, Jasper
- Physical Review X, Vol. 7, Issue 4
Topological Hopf-Chern insulators and the Hopf superconductor
journal, July 2016
- Kennedy, Ricardo
- Physical Review B, Vol. 94, Issue 3
Periodic table for topological bands with non-Hermitian symmetries
journal, June 2019
- Zhou, Hengyun; Lee, Jong Yeon
- Physical Review B, Vol. 99, Issue 23
Exceptional points in a non-Hermitian topological pump
journal, May 2017
- Hu, Wenchao; Wang, Hailong; Shum, Perry Ping
- Physical Review B, Vol. 95, Issue 18
Wannier representation of topological insulators
journal, January 2011
- Soluyanov, Alexey A.; Vanderbilt, David
- Physical Review B, Vol. 83, Issue 3
Z2Pack: Numerical implementation of hybrid Wannier centers for identifying topological materials
journal, February 2017
- Gresch, Dominik; Autès, Gabriel; Yazyev, Oleg V.
- Physical Review B, Vol. 95, Issue 7
Topology and data
journal, January 2009
- Carlsson, Gunnar
- Bulletin of the American Mathematical Society, Vol. 46, Issue 2
Non-Hermitian perspective of the band structure in heavy-fermion systems
journal, July 2018
- Yoshida, Tsuneya; Peters, Robert; Kawakami, Norio
- Physical Review B, Vol. 98, Issue 3
Berry phase effects on electronic properties
journal, July 2010
- Xiao, Di; Chang, Ming-Che; Niu, Qian
- Reviews of Modern Physics, Vol. 82, Issue 3
Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
journal, October 2017
- Bzdušek, Tomáš; Sigrist, Manfred
- Physical Review B, Vol. 96, Issue 15
All Magic Angles in Twisted Bilayer Graphene are Topological
journal, July 2019
- Song, Zhida; Wang, Zhijun; Shi, Wujun
- Physical Review Letters, Vol. 123, Issue 3
Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics
journal, November 2019
- Lee, Jong Yeon; Ahn, Junyeong; Zhou, Hengyun
- Physical Review Letters, Vol. 123, Issue 20
Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates
journal, May 2011
- Wan, Xiangang; Turner, Ari M.; Vishwanath, Ashvin
- Physical Review B, Vol. 83, Issue 20