This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls. Augmenting a given Hamiltonian by one or several domain walls results in confinement that naturally yields a Fredholm operator, whose index is taken as the topological charge of the system. The index is computed explicitly in terms of the symbol of the Hamiltonian by a Fedosov–Hörmander formula, which implements in Euclidean spaces an Atiyah–Singer index theorem. For Hamiltonians admitting an appropriate decomposition in a Clifford algebra, the index is given by the easily computable topological degree of a naturally associated map. A practically important property of topological insulators is the asymmetric transport observed along one-dimensional lines generated by the domain walls. This asymmetry is captured by the edge conductivity, a physical observable of the system. We prove that the edge conductivity is quantized and given by the index of a second Fredholm operator of the Toeplitz type. We also prove topological charge conservation by stating that the two aforementioned indices agree. This result generalizes to higher dimensions and higher-order topological insulators, the bulk-edge correspondence of two-dimensional materials. We apply this procedure to evaluate the topological charge of several classical examples of (standard and higher-order) topological insulators and superconductors in one, two, and three spatial dimensions.
REFERENCES
1.
Avron
, J. E.
, Seiler
, R.
, and Simon
, B.
, “Quantum Hall effect and the relative index for projections
,” Phys. Rev. Lett.
65
, 2185
–2188
(1990
).2.
Avron
, J. E.
, Seiler
, R.
, and Simon
, B.
, “Charge deficiency, charge transport and comparison of dimensions
,” Commun. Math. Phys.
159
, 399
–422
(1994
).3.
Bal
, G.
, “Continuous bulk and interface description of topological insulators
,” J. Math. Phys.
60
, 081506
(2019
).4.
Bal
, G.
, “Topological protection of perturbed edge states
,” Commun. Math. Sci.
17
, 193
–225
(2019
).5.
6.
Bal
, G.
, “Topological invariants for interface modes
,” Commun. Partial Differ. Equations
47
(8
), 1636
–1679
(2022
).7.
Bal
, G.
, Becker
, S.
, and Drouot
, A.
, “Magnetic slowdown of topological edge states
,” Commun. Pure Appl. Math.
(to be published) (2023
); arXiv:2201.07133.8.
Bal
, G.
, Becker
, S.
, Drouot
, A.
, Kammerer
, C. F.
, Lu
, J.
, and Watson
, A.
, “Edge state dynamics along curved interfaces
,” SIAM J. Math. Anal.
(to be published) (2023
); arXiv:2106.00729.9.
Bal
, G.
and Massatt
, D.
, “Multiscale invariants of Floquet topological insulators
,” Multiscale Model. Simul.
20
, 493
–523
(2022
).10.
Bellissard
, J.
, van Elst
, A.
, and Schulz‐Baldes
, H.
, “The noncommutative geometry of the quantum Hall effect
,” J. Math. Phys.
35
, 5373
–5451
(1994
).11.
Bernevig
, B. A.
and Hughes
, T. L.
, Topological Insulators and Topological Superconductors
(Princeton University Press
, 2013
).12.
Bleecker
, D.
and Booss
, B.
, Index Theory with Applications to Mathematics and Physics
(International Press
, 2013
).13.
Bolte
, J.
and Glaser
, R.
, “A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators
,” Commun. Math. Phys.
247
, 391
–419
(2004
).14.
Bony
, J.-M.
, “Caractérisations des opérateurs pseudo-différentiels
,” in Séminaire Équations aux Dérivées Partielles (Polytechnique)
(Cedram
, 1996
), pp. 1
–15
.15.
Bony
, J.-M.
, “On the characterization of pseudodifferential operators (old and new)
,” in Studies in Phase Space Analysis with Applications to PDEs
(Springer
, 2013
), pp. 21
–34
.16.
Bony
, J.-M.
and Chemin
, J.-Y.
, “Espaces fonctionnels associés au calcul de Weyl-Hörmander
,” Bull. Soc. Math. Fr.
122
, 77
–118
(1994
).17.
Bony
, J.-M.
and Lerner
, N.
, “Quantification asymptotique et microlocalisations d’ordre supérieur. I
,” Ann. Sci. Ec. Norm. Super.
22
, 377
–433
(1989
).18.
Bott
, R.
and Seeley
, R.
, “Some remarks on the paper of Callias
,” Commun. Math. Phys.
62
, 235
–245
(1978
).19.
Bourne
, C.
, Kellendonk
, J.
, and Rennie
, A.
, “The K-theoretic bulk–edge correspondence for topological insulators
,” Ann. Henri Poincare
18
, 1833
–1866
(2017
).20.
Bourne
, C.
and Rennie
, A.
, “Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
,” Math. Phys., Anal. Geom.
21
, 16
(2018
).21.
Callias
, C.
, “Axial anomalies and index theorems on open spaces
,” Commun. Math. Phys.
62
, 213
–234
(1978
).22.
Combes
, J.-M.
and Germinet
, F.
, “Edge and impurity effects on quantization of Hall currents
,” Commun. Math. Phys.
256
, 159
–180
(2005
).23.
Davies
, E. B.
, Spectral Theory and Differential Operators
, Cambridge Studies in Advanced Mathematics (Cambridge University Press
, 1995
).24.
Delplace
, P.
, Marston
, J. B.
, and Venaille
, A.
, “Topological origin of equatorial waves
,” Science
358
, 1075
–1077
(2017
).25.
Dimassi
, M.
and Sjöstrand
, J.
, Spectral Asymptotics in the Semi-Classical Limit
(Cambridge University Press
, 1999
), Vol. 268.26.
Dombrowski
, N.
, Germinet
, F.
, and Raikov
, G.
, “Quantization of edge currents along magnetic barriers and magnetic guides
,” Ann. Henri Poincare
12
, 1169
–1197
(2011
).27.
Drouot
, A.
, “Microlocal analysis of the bulk-edge correspondence
,” Commun. Math. Phys.
383
, 2069
–2112
(2021
).28.
Dubrovin
, B. A.
, Fomenko
, A. T.
, and Novikov
, S. P.
, Modern Geometry—Methods and Applications. Part II: The Geometry and Topology of Manifolds
(Springer-Verlag
, New York
, 1985
).29.
Elbau
, P.
and Graf
, G. M.
, “Equality of bulk and edge Hall conductance revisited
,” Commun. Math. Phys.
229
, 415
–432
(2002
).30.
Essin
, A. M.
and Gurarie
, V.
, “Bulk-boundary correspondence of topological insulators from their respective Green’s functions
,” Phys. Rev. B
84
, 125132
(2011
).31.
Fedosov
, B. V.
, “Direct proof of the formula for the index of an elliptic system in Euclidean space
,” Funct. Anal. Appl.
4
, 339
–341
(1970
).32.
Fefferman
, C. L.
, Lee-Thorp
, J. P.
, and Weinstein
, M. I.
, “Edge states in honeycomb structures
,” Ann. PDE
2
, 12
(2016
).33.
Fefferman
, C. L.
and Weinstein
, M. I.
, “Honeycomb lattice potentials and Dirac points
,” J. Am. Math. Soc.
25
, 1169
–1220
(2012
).34.
Fefferman
, C. L.
and Weinstein
, M. I.
, “Wave packets in honeycomb structures and two-dimensional Dirac equations
,” Commun. Math. Phys.
326
, 251
–286
(2014
).35.
Fukui
, T.
, Shiozaki
, K.
, Fujiwara
, T.
, and Fujimoto
, S.
, “Bulk-edge correspondence for Chern topological phases: A viewpoint from a generalized index theorem
,” J. Phys. Soc. Jpn.
81
, 114602
(2012
).36.
Graf
, G. M.
, “Aspects of the integer quantum Hall effect
,” in Proceedings of Symposia in Pure Mathematics
(American Mathematical Society
, Providence, RI
, 1998; 2007
), Vol. 76, p. 429
.37.
Graf
, G. M.
, Jud
, H.
, and Tauber
, C.
, “Topology in shallow-water waves: A violation of bulk-edge correspondence
,” Commun. Math. Phys.
383
, 731
–761
(2021
).38.
Graf
, G. M.
and Porta
, M.
, “Bulk-edge correspondence for two-dimensional topological insulators
,” Commun. Math. Phys.
324
, 851
–895
(2013
).39.
Hatsugai
, Y.
, “Chern number and edge states in the integer quantum Hall effect
,” Phys. Rev. Lett.
71
, 3697
(1993
).40.
Hörmander
, L. V.
, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators
(Springer-Verlag
, 1994
).41.
Kellendonk
, J.
and Schulz-Baldes
, H.
, “Quantization of edge currents for continuous magnetic operators
,” J. Funct. Anal.
209
, 388
–413
(2004
).42.
43.
Liu
, C.-X.
, Qi
, X.-L.
, Zhang
, H.
, Dai
, X.
, Fang
, Z.
, and Zhang
, S.-C.
, “Model Hamiltonian for topological insulators
,” Phys. Rev. B
82
, 045122
(2010
).44.
Ludewig
, M.
and Thiang
, G. C.
, “Cobordism invariance of topological edge-following states
,” arXiv:2001.08339 (2020
).45.
Nicola
, F.
and Rodino
, L.
, Global Pseudo-Differential Calculus on Euclidean Spaces
(Springer Science & Business Media
, 2011
), Vol. 4.46.
Nirenberg
, L.
, Topics in Nonlinear Functional Analysis
(American Mathematical Society
, 1974
), Vol. 6.47.
Prodan
, E.
and Schulz-Baldes
, H.
, Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
(Springer-Verlag
, Berlin
, 2016
).48.
Qi
, X.-L.
and Zhang
, S.-C.
, “Topological insulators and superconductors
,” Rev. Mod. Phys.
83
, 1057
–1110
(2011
).49.
Quinn
, S.
and Bal
, G.
, “Approximations of interface topological invariants
,” arXiv:2112.02686 (2022
).50.
Quinn
, S.
and Bal
, G.
, “Asymmetric transport for magnetic Dirac equations
,” arXiv:2211.00726 (2022
).51.
Sato
, M.
and Ando
, Y.
, “Topological superconductors: A review
,” Rep. Prog. Phys.
80
, 076501
(2017
).52.
Schindler
, F.
, Cook
, A. M.
, Vergniory
, M. G.
, Wang
, Z.
, Parkin
, S. S. P.
, Bernevig
, B. A.
, and Neupert
, T.
, “Higher-order topological insulators
,” Sci. Adv.
4
, eaat0346
(2018
).53.
Schulz-Baldes
, H.
, Kellendonk
, J.
, and Richter
, T.
, “Simultaneous quantization of edge and bulk Hall conductivity
,” J. Phys. A: Math. Gen.
33
, L27
(2000
).54.
Souslov
, A.
, Dasbiswas
, K.
, Fruchart
, M.
, Vaikuntanathan
, S.
, and Vitelli
, V.
, “Topological waves in fluids with odd viscosity
,” Phys. Rev. Lett.
122
, 128001
(2019
).55.
Tauber
, C.
, Delplace
, P.
, and Venaille
, A.
, “A bulk-interface correspondence for equatorial waves
,” J. Fluid Mech.
868
, R2
(2019
).56.
Volovik
, G.
, The Universe in a Helium Droplet
, International Series of Monographs on Physics (OUP
, Oxford
, 2009
).57.
Witten
, E.
, “Three lectures on topological phases of matter
,” Riv. Nuovo Cimento
39
, 313
–370
(2016
).58.
Zworski
, M.
, Semiclassical Analysis
(American Mathematical Society
, 2012
), Vol. 138.59.
We use Δ for the order parameter as is customary in the superconductor literature. The (positive) Laplace operator is denoted by D · D.
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