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Strongly disordered Floquet topological systems. (English) Zbl 07063405
Summary: We study the strong disorder regime of Floquet topological systems in dimension two that describe independent electrons on a lattice subject to a periodic driving. In the spectrum of the Floquet propagator we assume the existence of an interval in which all states are localized – a mobility gap – extending previous studies which make the stronger spectral gap assumption. We devise a new approach to define the topological invariants by way of stretching the gap of a given system onto the whole circle. We show that such completely localized systems have natural indices that circumvent the relative construction and match with quantized magnetization and pumping observables from the physics literature. These indices obey a bulk-edge correspondence, which carries over to the stretched systems as well. Finally, these invariants are shown to coincide with those associated with the usual relative construction, which we also extend to the mobility gap regime.

60K Special processes
82B Equilibrium statistical mechanics
47A General theory of linear operators
47B Special classes of linear operators
Full Text: DOI
[1] Aizenman, M.; Graf, GM, Localization bounds for an electron gas, J. Phys. A Math. Gen., 31, 6783-6806, (1998) · Zbl 0953.82009
[2] Aizenman, M., Warzel, S.: Random Operators. Amer. Math. Soc. (2015) · Zbl 1333.82001
[3] Altland, A.; Zirnbauer, MR, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B., 55, 1142-1161, (1997)
[4] Asch, J., Bourget, O., Joye, A.: Chirality induced interface currents in the Chalker Coddington model. arXiv preprint. arXiv:1708.02120 (2017) · Zbl 1210.82033
[5] Asch, J.; Bourget, O.; Joye, A., Dynamical localization of the Chalker-Coddington model far from transition, J. Stat. Phys., 147, 194-205, (2012) · Zbl 1243.82036
[6] Carpentier, D.; etal., Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals, Nucl. Phys. B, 896, 779-834, (2015) · Zbl 1331.82065
[7] Combes, JM; Thomas, L., Asymptotic behaviour of eigenfunctions for multiparticle Schrodinger operators, Commun. Math. Phys., 34, 251-270, (1973) · Zbl 0271.35062
[8] Delplace, P.; Fruchart, M.; Tauber, C., Phase rotation symmetry and the topology of oriented scattering networks, Phys. Rev. B, 95, 205413, (2017)
[9] Elbau, P.; Graf, GM, Equality of bulk and edge Hall conductance revisited, Commun. Math. Phys., 229, 415-432, (2002) · Zbl 1001.81091
[10] Elgart, A.; Graf, GM; Schenker, J., Equality of the bulk and edge Hall conductances in a mobility gap, Commun. Math. Phys., 259, 185-221, (2005) · Zbl 1086.81081
[11] Enss, V.; Veselic, K., Bound states and propagating states for time-dependent Hamiltonians, Ann. de l’l.H.P. Phys. Theorique., 39, 159-191, (1983) · Zbl 0532.47007
[12] Fruchart, M.; etal., Probing (topological) Floquet states through DC transport, Physica E Low Dimens. Syst. Nanostruct., 75, 287-294, (2016)
[13] Fruchart, M., Complex classes of periodically driven topological lattice systems, Phys. Rev. B, 93, 115429, (2016)
[14] Fulga, IC; Maksymenko, M., Scattering matrix invariants of Floquet topological insulators, Phys. Rev. B, 93, 075405, (2016)
[15] Graf, GM; Shapiro, J., The bulk-edge correspondence for disordered chiral chains, Commun. Math. Phys., 363, 829-846, (2018) · Zbl 1401.82031
[16] Graf, GM; Tauber, C., Bulk-edge correspondence for two-dimensional Floquet topological insulators, Ann. Henri Poincare., 19, 709-741, (2018) · Zbl 1392.82008
[17] Hamza, E.; Joye, A.; Stolz, G., Dynamical localization for unitary Anderson models, Math. Phys. Anal. Geom., 12, 381, (2009) · Zbl 1186.82045
[18] Hunziker, W.; Sigal, IM, The quantum N-body problem, J. Math. Phys., 41, 3448-3510, (2000) · Zbl 0981.81026
[19] Joye, A., Dynamical localization for d-dimensional random quantum walks, Quantum. Inf. Process., 11, 1251-1269, (2012) · Zbl 1252.82087
[20] Kundu, A.; Fertig, HA; Seradjeh, B., Effective theory of Floquet topological transitions, Phys. Rev. Lett., 113, 236803, (2014)
[21] Mbarek, A.: Helffer-Sjostrand Formula for Unitary Operators. arXiv preprint. arXiv:1506.04537 (2015)
[22] Nathan, F.; etal., Quantized magnetization density in periodically driven systems, Phys. Rev. Lett., 119, 186801, (2017)
[23] Oka, T.; Aoki, H., Photovoltaic Hall effect in graphene, Phys. Rev. B, 79, 081406, (2009)
[24] Prodan, E.; Schulz-Baldes, H., Non-commutative odd Chern numbers and topological phases of disordered chiral systems, J. Funct. Anal., 271, 1150-1176, (2016) · Zbl 1344.82055
[25] Quelle, A., et al.: Driving protocol for a Floquet topological phase without static counterpart. New J. Phys. 19, (2017)
[26] Rio, R.; etal., Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, Journal d’Analyse Mathematique, 69, 153200, (1996)
[27] Rodriguez-Vega, M.; Fertig, HA; Seradjeh, B., Quantum noise detects Floquet topological phases, Phys. Rev. B, 98, 041113, (2018)
[28] Roy, R.; Harper, F., Periodic table for Floquet topological insulators, Phys. Rev. B, 96, 155118, (2017)
[29] Rudner, MS; etal., Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X, 3, 031005, (2013)
[30] Sadel, C.; Schulz-Baldes, H., Topological boundary invariants for Floquet systems and quantum walks, Math. Phys. Anal. Geom., 20, 22, (2017) · Zbl 1413.46061
[31] Simon, B., Cyclic vectors in the Anderson model, Rev. Math. Phys., 06, 1183-1185, (1994) · Zbl 0841.60081
[32] Tauber, C., Effective vacua for Floquet topological phases: a numerical perspective on the switch-function formalism, Phys. Rev. B, 97, 195312, (2018)
[33] Tauber, C.; Delplace, P., Topological edge states in two-gap unitary systems: a transfer matrix approach, New J. Phys., 17, 115008, (2015)
[34] Titum, P.; etal., Anomalous Floquet-Anderson insulator as a nonadiabatic quantized charge pump, Phys. Rev. X, 6, 021013, (2016)
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