A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.
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Asbóth, J.K., Tarasinski, B., Delplace, P.: Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems. Phys. Rev. B 90, 125143 (2014)
Asch, J., Bourget, O., Joye, A.: Spectral stability of unitary network models. Rev. Math. Phys. 27, 1530004 (2015)
Asch, J., Bourget, O., Joye, A.: Chirality induced interface currents in the Chalker Coddington model. arXiv:1708.02120
Bellissard, J.: K-theory of C ∗-algebras in solid state physics. In: Dorlas, T., Hugenholtz, M., Winnink, M. (eds.) Statistical mechanics and field theory: mathematical aspects, lecture notes in physics, vol. 257, pp 99–156. Springer, Berlin (1986)
Carpentier, D., Delplace, P., Fruchart, M., Gawedzki, K., Tauber, C.: Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals. Nucl. Phys. B 896, 779–834 (2015)
Chalker, J.T., Coddington, P.D.: Percolation, quantum tunnelling and the integer Hall effect. J. Phys. C Solid State 21, 2665 (1988)
Delplace, P., Fruchart, M., Tauber, C.: Phase rotation symmetry and the topology of oriented scattering networks. Phys. Rev. B 95, 205413 (2017)
Elbau, P., Graf, G.M.: Equality of bulk and edge Hall conductance revisited. Commun. Math. Phys. 229, 415–432 (2002)
Fruchart, M.: Complex classes of periodically driven topological lattice systems. Phys. Rev. B 93, 115429 (2016)
Graf, G.M., Tauber, C.: Bulk-Edge correspondence for two-dimensional Floquet topological insulators. arXiv:1707.09212
Ho, C.-M., Chalker, J.T.: Models for the integer quantum Hall effect: The network model, the Dirac equation, and a tight-binding Hamiltonian. Phys. Rev. B 54, 8708 (1996)
Nathan, F., Rudner, M.S.: Topological singularities and the general classification of Floquet-Bloch systems. New J Phys. 17, 125014 (2015)
Kellendonk, J., Richter, T., Schulz-Baldes, H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14, 87–119 (2002)
Kitagawa, T.: Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quantum Inf. Process 11(5), 1107–1148 (2012)
Pasek, M., Chong, Y.D.: Network models of photonic Floquet topological insulators. Phys. Rev. B 89, 075113 (2014)
Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From K-theory to Physics. Springer International Publishing, Szwitzerland (2016)
Rordam, M., Larsen, F., Laustsen, N.: An Introduction to K-theory for C ∗-Algebras. Cambridge University Press, Cambridge (2000)
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)
Ryu, S., Schnyder, A.P., Furusaki, A., Ludwig, A.W.W.: Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010)
Schulz-Baldes, H., Kellendonk, J., Richter, T.: Similtaneous quantization of edge and bulk Hall conductivity. J. Phys. A 33, L27–L32 (2000)
Wegge-Olsen, N.E.: K-Theory and C ∗-Algebras. Oxford University Press, Oxford (1993)
The authors thank Rafeal Tiedra for discussions on quantum walks, and the Instituto de Matematicas (UNAM), Cuernavaca, for its hospitality during a visit in July 2017 when this work was written. We also thank an unknown referee for a careful reading and constructive comments. An independent contribution on similar matters by Graf and Tauber  appeared on the arXiv while preparing the final version of this manuscript. It also proves bulk-boundary correspondence for Floquet systems, but is restricted to two-dimensional models. It provides a more ad hoc functional analytic treatment, while here the general theory of bulk-boundary correspondence of topological insulators from  is combined with a K-theoretic fact (Theorem 2) not contained in . Before preparing the final revision of this work, another related draft on edge states in the Chalker-Coddington model was posted by Asch, Bourget and Joye . This research was partly supported by the Chilean grant FONDECYT Regular 1161651 and the DFG.
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Sadel, C., Schulz-Baldes, H. Topological Boundary Invariants for Floquet Systems and Quantum Walks. Math Phys Anal Geom 20, 22 (2017). https://doi.org/10.1007/s11040-017-9253-1
- Floquet topological insulators
- Quantum walks
- Connecting maps in K-theory
Mathematics Subject Classifications (2010)