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Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals. (English) Zbl 1331.82065
Summary: We present mathematical details of the construction of a topological invariant for periodically driven two-dimensional lattice systems with time-reversal symmetry and quasienergy gaps, which was proposed recently by some of us. The invariant is represented by a gap-dependent \(\mathbb{Z}_2\)-valued index that is simply related to the Kane-Mele invariants of quasienergy bands but contains an extra information. As a byproduct, we prove new expressions for the two-dimensional Kane-Mele invariant relating the latter to Wess-Zumino amplitudes and the boundary gauge anomaly.

82D25 Statistical mechanical studies of crystals
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[1] Albert, V. V.; Glazman, L. I.; Jiang, L., Topological properties of linear circuit lattices, (2014)
[2] Bellec, M.; Kuhl, U.; Montambaux, G.; Mortessagne, F., Manipulation of edge states in microwave artificial graphene, New J. Phys., 16, 11, 113023, (2014)
[3] Bott, R.; Seeley, J. R., Some remarks on the paper of callias: “axial anomalies and index theorems on open spaces”, Commun. Math. Phys., 62, 235-245, (1978) · Zbl 0409.58019
[4] Bott, R.; Tu, L. W., Differential forms in algebraic topology, (1982), Springer · Zbl 0496.55001
[5] Carpentier, D.; Delplace, P.; Fruchart, M.; Gawędzki, K., Topological index for periodically driven time-reversal invariant 2d systems, Phys. Rev. Lett., 114, 106806, (2015) · Zbl 1331.82065
[6] Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P., Modern geometry - methods and applications: part II: the geometry and topology of manifolds, (1985), Springer · Zbl 0565.57001
[7] De Nittis, G.; Gomi, K., Classification of “quaternionic” Bloch-bundles: topological quantum systems of type AII, (2014)
[8] Fruchart, M.; Carpentier, D., An introduction to topological insulators, C. R. Phys., 14, 779-815, (2013)
[9] Fu, L.; Kane, C. L., Time reversal polarization and a \(Z_2\) adiabatic spin pump, Phys. Rev. B, 74, 195312, (2006)
[10] Fu, L.; Kane, C. L., Topological insulators with inversion symmetry, Phys. Rev. B, 76, 045302, (2007)
[11] Freed, D. S.; Moore, G. W., Twisted equivariant matter, Ann. Henri Poincaré, 14, 1927-2023, (2013) · Zbl 1286.81109
[12] Fiorenza, D.; Monaco, D.; Panati, G., \(Z_2\) invariants of topological insulators as geometric obstructions, (2014)
[13] Fox, R. H., Homotopy groups and torus homotopy groups, Ann. Math., 49, 471-510, (1948) · Zbl 0038.36602
[14] Gawȩdzki, K., Topological actions in two-dimensional quantum field theory, (’t Hooft, G.; Jaffe, A.; Mack, G.; Mitter, P.; Stora, R., Nonperturbative Quantum Field Theory, (1988), Plenum Press), 101-141
[15] Gawȩdzki, K., Conformal field theory: a case study, (Nutku, Y.; Saclioglu, C.; Turgut, T., New Non-Perturbative Methods in String and Field Theory, (2000), Perseus Publishing), 1-55
[16] Gawȩdzki, K.; Reis, N., WZW branes and gerbes, Rev. Math. Phys., 14, 1281-1334, (2002) · Zbl 1033.81067
[17] Gawȩdzki, K.; Suszek, R. R.; Waldorf, K., Bundle gerbes for orientifold sigma models, Adv. Theor. Math. Phys., 15, 621-688, (2011) · Zbl 1280.81089
[18] Hafezi, M.; Demler, E. A.; Lukin, M. D.; Taylor, J. M., Robust optical delay lines with topological protection, Nat. Phys., 7, 907-912, (2011)
[19] Hitchin, Nigel J., What is ... a gerbe?, Not. Am. Math. Soc., 50, 218-219, (2003) · Zbl 1298.53001
[20] Hafezi, M.; Mittal, S.; Fan, J.; Migdall, A.; Taylor, J. M., Imaging topological edge states in silicon photonics, Nat. Photonics, 7, 1001-1005, (2013)
[21] Hu, W.; Pillay, J. C.; Wu, K.; Pasek, M.; Shum, P. P.; Chong, Y. D., Measurement of a topological edge invariant in a microwave network, Phys. Rev. X, 5, 1, 011012, (2015)
[22] Haldane, F. D.M.; Raghu, S., Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett., 100, 013904, (2008)
[23] Husemoller, D., Fibre bundles, (1993), Springer · Zbl 0794.55001
[24] Inoue, J.-I.; Tanaka, A., Photoinduced transition between conventional and topological insulators in two-dimensional electronic systems, Phys. Rev. Lett., 105, 1, 017401, (2010)
[25] Jotzu, G.; Messer, M.; Desbuquois, R.; Lebrat, M.; Uehlinger, T.; Greif, D.; Esslinger, T., Experimental realization of the topological Haldane model with ultracold fermions, Nature, 515, 237-240, (2014)
[26] Jia, N.; Sommer, A.; Schuster, D.; Simon, J., Time reversal invariant topologically insulating circuits, (2013)
[27] Kitagawa, T.; Broome, M. A.; Fedrizzi, A.; Rudner, M. S.; Berg, E.; Kassal, I.; Aspuru-Guzik, A.; Demler, E.; White, A. G., Observation of topologically protected bound states in photonic quantum walks, Nat. Commun., 3, 882, (2012)
[28] Kitagawa, T.; Berg, E.; Rudner, M.; Demler, E., Topological characterization of periodically driven quantum systems, Phys. Rev. B, 82, 23, 235114, (2010)
[29] Klitzing, K.v.; Dorda, G.; Pepper, M., New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett., 45, 494-497, (1980)
[30] Kitaev, A., Periodic table for topological insulators and superconductors, AIP Conf. Proc., 1134, 1, 22-30, (2009) · Zbl 1180.82221
[31] Kane, C. L.; Lubensky, T. C., Topological boundary modes in isostatic lattices, Nat. Phys., 10, 39-45, (2014)
[32] Kane, C. L.; Mele, E. J., \(Z_2\) topological order and the quantum spin Hall effect, Phys. Rev. Lett., 95, 146802, (2005)
[33] Kitagawa, T.; Oka, T.; Brataas, A.; Fu, L.; Demler, E., Transport properties of nonequilibrium systems under the application of light: photoinduced quantum Hall insulators without Landau levels, Phys. Rev. B, 84, 23, 235108, (2011)
[34] Lee, S.-S.; Ryu, S., Many-body generalization of the \(Z_2\) topological invariant for the quantum spin Hall effect, Phys. Rev. Lett., 100, 186807, (2008)
[35] Lindner, N. H.; Refael, G.; Galitski, V., Floquet topological insulator in semiconductor quantum wells, Nat. Phys., 7, 490-495, (2011)
[36] Moore, J. E.; Balents, L., Topological invariants of time-reversal-invariant band structures, Phys. Rev. B, 75, 121-306, (2007)
[37] Meinrenken, E., The basic gerbe over a compact simple Lie group, Enseign. Math., 49, 307-333, (2003) · Zbl 1061.53034
[38] Milnor, J. W., Topology from the differentiable viewpoint, (1965), The University of Virginia Press · Zbl 0136.20402
[39] Manton, N.; Sutcliffe, P., Topological solitons, (2004), Cambridge University Press · Zbl 1100.37044
[40] Oka, T.; Aoki, H., Photovoltaic Hall effect in graphene, Phys. Rev. B, 79, 8, 081406, (2009)
[41] Panati, G., Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré, 8, 995-1011, (2007) · Zbl 1375.81102
[42] Polyakov, A.; Wiegmann, P. B., Theory of nonabelian Goldstone bosons in two dimensions, Phys. Lett. B, 131, 121-126, (1983)
[43] Qi, X.-L.; Hughes, T. L.; Zhang, S.-C., Topological field theory of time-reversal invariant insulators, Phys. Rev. B, 78, 195424, (2008)
[44] Rudner, M. S.; Lindner, N. H.; Berg, E.; Levin, M., Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X, 3, 031005, (2013)
[45] Ryu, S.; Schnyder, A. P.; Furusaki, A.; Ludwig, A. W.W., Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys., 12, 6, 065010, (2010)
[46] Rechtsman, M. C.; Zeuner, J. M.; Plotnik, Y.; Lumer, Y.; Podolsky, D.; Dreisow, F.; Nolte, S.; Segev, M.; Szameit, A., Photonic Floquet topological insulators, Nature, 496, 196-200, (April 2013)
[47] Schnyder, A. P.; Ryu, S.; Furusaki, A.; Ludwig, A. W.W., Classification of topological insulators and superconductors, AIP Conf. Proc., 1134, 1, 10-21, (2009) · Zbl 1180.82228
[48] Schreiber, U.; Sweigert, C.; Waldorf, K., Unoriented WZW models and holonomy of bundle gerbes, Commun. Math. Phys., 274, 31-64, (2007) · Zbl 1148.53057
[49] Witten, E., Non-abelian bosonization in two dimensions, Commun. Math. Phys., 92, 455-472, (1984) · Zbl 0536.58012
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