Short-ranged and line-gapped non-Hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral localizer. This numerical technique is implemented in an example with relevance to the design of topological photonic systems, such as topological lasers.
REFERENCES
1.
T.
Loring
and H.
Schulz-Baldes
, “Finite volume calculation of K-theory invariants
,” N. Y. J. Math.
22
, 1111
–1140
(2017
), https://nyjm.albany.edu/j/2017/23-48.html.2.
T.
Loring
and H.
Schulz-Baldes
, “The spectral localizer for even index pairings
,” J. Noncommutative Geom.
14
, 1
–23
(2020
).3.
H.
Schulz-Baldes
and T.
Stoiber
, “The spectral localizer for semifinite spectral triples
,” Proc. Am. Math. Soc.
149
, 121
–134
(2020
).4.
N.
Doll
and H.
Schulz-Baldes
, “Approximate symmetries and conservation laws in topological insulators and associated -invariants
,” Ann. Phys.
419
, 168238
(2020
).5.
N.
Doll
and H.
Schulz-Baldes
, “Skew localizer and -flows for real index pairings
,” Adv. Math.
392
, 108038
(2021
).6.
H.
Schulz-Baldes
and T.
Stoiber
, “Invariants of disordered semimetals via the spectral localizer
,” Europhys. Lett.
136
, 027001
(2021
).7.
A.
Cerjan
and T. A.
Loring
, “Local invariants identify topology in metals and gapless systems
,” Phys. Rev. B
106
, 064109
(2022
).8.
W.
Cheng
, A.
Cerjan
, S.-Y.
Chen
, E.
Prodan
, T. A.
Loring
, and C.
Prodan
, “Revealing topology in metals using experimental protocols inspired by K-theory
,” Nat. Commun.
14
, 3071
(2023
).9.
H.
Liu
and I. C.
Fulga
, “Mixed higher-order topology: Boundary non-hermitian skin effect induced by a Floquet bulk
,” Phys. Rev. B
108
, 035107
(5 July 2022
).10.
T.
Ozawa
, H. M.
Price
, A.
Amo
, N.
Goldman
, M.
Hafezi
, L.
Lu
, M. C.
Rechtsman
, D.
Schuster
, J.
Simon
, O.
Zilberberg
, and I.
Carusotto
, “Topological photonics
,” Rev. Mod. Phys.
91
, 015006
(2019
).11.
K.
Kawabata
, K.
Shiozaki
, M.
Ueda
, and M.
Sato
, “Symmetry and topology in non-hermitian physics
,” Phys. Rev. X
9
, 041015
(2019
).12.
G.
De Nittis
and M.
Lein
, “The Schrödinger formalism of electromagnetism and other classical waves—How to make quantum-wave analogies rigorous
,” Ann. Phys.
396
, 579
–617
(2018
).13.
E. J.
Bergholtz
, J. C.
Budich
, and F. K.
Kunst
, “Exceptional topology of non-Hermitian systems
,” Rev. Mod. Phys.
93
, 015005
(2021
).14.
Y.
Ashida
, Z.
Gong
, and M.
Ueda
, “Non-hermitian physics
,” Adv. Phys.
69
, 249
–435
(2020
).15.
J. M.
Gracia-Bondía
, J. C.
Várilly
, and H.
Figueroa
, Elements of Noncommutative Geometry
(Birkhäuser
, Boston
, 2001
).16.
N.
Doll
, H.
Schulz-Baldes
, and N.
Waterstraat
, (De Gruyter
, 2023
).17.
E.
Prodan
and H.
Schulz-Baldes
, Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
(Springer International
, Cham
, 2016
).18.
V.
Peano
and H.
Schulz-Baldes
, “Topological edge states for disordered bosonic systems
,” J. Math. Phys.
59
, 031901
(2018
).19.
J.
Kaad
, “On the unbounded picture of KK-theory
,” SIGMA
16
, 082
(2020
).20.
H.
Schulz-Baldes
and T.
Stoiber
, “Callias-type operators associated to spectral triples
,” J. Noncommutative Geom.
17
(2
), 527
–572
(2023
).21.
F. D. M.
Haldane
, “Model for a quantum Hall effect without Landau Levels: condensed-matter realization of the “‘parity anomaly
,’” Phys. Rev. Lett.
61
, 2015
(1988
).22.
A.
Cerjan
and T. A.
Loring
, “An operator-based approach to topological photonics
,” Nanophotonics
11
, 4765
(2022
).23.
A.
Taflove
and S. C.
Hagness
, Computational Electrodynamics: The Finite-Difference Time-Domain Method
(Artech House
, Boston
, 2005
).24.
W.
Li
and L. C.
Paulson
, “Evaluating winding numbers and counting complex roots through Cauchy indices in Isabelle/HOL
,” J. Autom. Reasoning
64
, 331
–360
(2020
).25.
R.
Shindou
, R.
Matsumoto
, S.
Murakami
, and J. I.
Ohe
, “Topological chiral magnonic edge mode in a magnonic crystal
,” Phys. Rev. B
87
, 174427
(2013
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
