We prove that a disordered analog of the Su–Schrieffer–Heeger model exhibits dynamical localization (i.e., the fractional moment condition) at all energies except possibly zero energy, which is singled out by chiral symmetry. Localization occurs at arbitrarily weak disorder, provided it is sufficiently random. If furthermore the hopping probability measures are properly tuned so that the zero energy Lyapunov spectrum does not contain zero, then the system exhibits localization also at that energy, which is of relevance for topological insulators. The method also applies to the usual Anderson model on the strip.
REFERENCES
1.
H.
Kunz
and B.
Souillard
, “Sur le spectre des opérateurs aux différences finies aléatoires
,” Commun. Math. Phys.
78
(2
), 201
–246
(1980
).2.
A.
Klein
, J.
Lacroix
, and A.
Speis
, “Localization for the Anderson model on a strip with singular potentials
,” J. Funct. Anal.
94
(1
), 135
–155
(1990
).3.
S.
Jitomirskaya
, H.
Schulz-Baldes
, and G.
Stolz
, “Delocalization in random polymer models
,” Commun. Math. Phys.
233
(1
), 27
–48
(2003
).4.
D.
Damanik
, R.
Sims
, and G.
Stolz
, “Localization for one-dimensional, continuum, Bernoulli-Anderson models
,” Duke Math. J.
114
(1
), 59
–100
(2002
).5.
H.
Boumaza
, “Localization for an Anderson-Bernoulli model with generic interaction potential
,” Tohoku Math. J.
65
(1
), 57
–74
(2013
).6.
G. M.
Graf
and J.
Shapiro
, “The bulk-edge correspondence for disordered chiral chains
,” Commun. Math. Phys.
363
(3
), 829
–846
(2018
).7.
A.
Altland
and M. R.
Zirnbauer
, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures
,” Phys. Rev. B
55
, 1142
–1161
(1997
).8.
M. Z.
Hasan
and C. L.
Kane
, “Colloquium: Topological insulators
,” Rev. Mod. Phys.
82
, 3045
–3067
(2010
).9.
W. P.
Su
, J. R.
Schrieffer
, and A. J.
Heeger
, “Solitons in polyacetylene
,” Phys. Rev. Lett.
42
, 1698
–1701
(1979
).10.
M.
Aizenman
and G. M.
Graf
, “Localization bounds for an electron gas
,” J. Phys. A: Math. Gen.
31
, 6783
–6806
(1998
).11.
D.
Damanik
and P.
Stollmann
, “Multi-scale analysis implies strong dynamical localization
,” Geom. Funct. Anal.
11
(1
), 11
–29
(2001
).12.
R.
Carmona
, A.
Klein
, and F.
Martinelli
, “Anderson localization for Bernoulli and other singular potentials
,” Commun. Math. Phys.
108
(1
), 41
–66
(1987
).13.
Y.
Guivarch
and A.
Raugi
, “Frontière de furstenberg, propriétés de contraction et théorèmes de convergence
,” Z. Wahrscheinlichkeitstheorie Verw. Geb.
69
(2
), 187
–242
(1985
).14.
J.
Chapman
and G.
Stolz
, “Localization for random block operators related to the XY spin chain
,” Ann. Henri Poincare
16
(2
), 405
–435
(2015
).15.
C.
Sadel
and H.
Schulz-Baldes
, “Random Dirac operators with time reversal symmetry
,” Commun. Math. Phys.
295
(1
), 209
–242
(2010
).16.
A. W. W.
Ludwig
, H.
Schulz-Baldes
, and M.
Stolz
, “Lyapunov spectra for all ten symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions
,” J. Stat. Phys.
152
(2
), 275
–304
(2013
).17.
J.
Asch
, O.
Bourget
, and A.
Joye
, “Localization properties of the Chalker–Coddington model
,” Ann. Henri Poincare
11
(7
), 1341
–1373
(2010
).18.
S.
Attal
, A.
Joye
, and C.-A.
Pillet
, Open Quantum Systems I. The Hamiltonian Approach
(Springer
, Berlin
, 2006
).19.
A.
Elgart
, G. M.
Graf
, and J.
Schenker
, “Equality of the bulk and edge Hall conductances in a mobility gap
,” Commun. Math. Phys.
261
(2
), 545
(2005
).20.
P. D.
Hislop
, “Lectures on random Schrödinger operators
,” in Fourth Summer School in Analysis and Mathematical Physics
, Contemporary Mathematics Vol. 476
(American Mathematical Society
, Providence, RI
, 2008
), pp. 41
–131
.21.
22.
P.
Bougerol
, Lacroix: Products of Random Matrices with Applications to Schrödinger Operators
, Progress in Probability
(Birkhäuser
, 2012
).23.
G. E.
Bredon
, Topology and Geometry
, Graduate Texts in Mathematics
(Springer
, 1993
).24.
B.
Simon
, “Localization in general one dimensional random systems, I. Jacobi matrices
,” Commun. Math. Phys.
102
(2
), 327
–336
(1985
).25.
I. Y.
Goldsheid
and G. A.
Margulis
, “Lyapunov indices of a product of random matrices
,” Russ. Math. Surv.
44
(5
), 11
(1989
).26.
R.
Carmona
and J.
Lacroix
, Spectral Theory of Random Schrödinger Operators
(Birkhäuser
, Boston
, 1990
).27.
G. M.
Graf
, “Anderson localization and the space-time characteristic of continuum states
,” J. Stat. Phys.
75
(1
), 337
–346
(1994
).28.
M.
Aizenman
, J. H.
Schenker
, R. M.
Friedrich
, and D.
Hundertmark
, “Finite-volume fractional-moment criteria for Anderson localization
,” Commun. Math. Phys.
224
(1
), 219
–253
(2001
).29.
J. M.
Combes
and L.
Thomas
, “Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators
,” Commun. Math. Phys.
34
, 251
–270
(1973
).30.
To be compatible with the rest of this paper, we denote a size of a matrix here by L instead of the usual N in the literature; the usual name of the conjecture is “the conjecture.”
31.
G.
Casati
, L.
Molinari
, and F.
Izrailev
, “Scaling properties of band random matrices
,” Phys. Rev. Lett.
64
, 1851
–1854
(1990
).32.
G.
Casati
, B. V.
Chirikov
, I.
Guarneri
, and F. M.
Izrailev
, “Band-random-matrix model for quantum localization in conservative systems
,” Phys. Rev. E
48
, R1613
–R1616
(1993
).33.
J.
Schenker
, “Eigenvector localization for random band matrices with power law band width
,” Commun. Math. Phys.
290
(3
), 1065
–1097
(2009
).34.
R.
Peled
, J.
Schenker
, M.
Shamis
, and S.
Sodin
, “On the Wegner orbital model
,” Int. Math. Res. Not.
2019
(4
), 1030
–1058
(2017
).35.
P.
Bourgade
, L.
Erdős
, H.-T.
Yau
, and J.
Yin
, “Universality for a class of random band matrices
,” Adv. Theor. Math. Phys.
21
(3
), 739
–800
(2017
).36.
P.
Bourgade
, H.-T.
Yau
, and J.
Yin
, “Random band matrices in the delocalized phase I: Quantum unique ergodicity and universality
,” Commun. Pure Appl. Math.
73
(7
), 1526
–1596
(2020
).37.
T.
Shcherbina
, “On the second mixed moment of the characteristic polynomials of 1D band matrices
,” Commun. Math. Phys.
328
(1
), 45
–82
(2014
).38.
M.
Shcherbina
and T.
Shcherbina
, “Characteristic polynomials for 1D random band matrices from the localization side
,” Commun. Math. Phys.
351
(3
), 1009
–1044
(2017
).39.
C. M.
Newman
, “The distribution of Lyapunov exponents: Exact results for random matrices
,” Commun. Math. Phys.
103
(1
), 121
–126
(1986
).40.
J.
Shapiro
and C.
Tauber
, “Strongly disordered Floquet topological systems
,” Ann. Henri Poincare
20
(6
), 1837
–1875
(2019
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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