zbMATH — the first resource for mathematics

Topological boundary invariants for Floquet systems and quantum walks. (English) Zbl 1413.46061
Summary: 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 depend only 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.

46L80 \(K\)-theory and operator algebras (including cyclic theory)
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
Full Text: DOI
[1] Asbóth, JK; Tarasinski, B; Delplace, P, Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems, Phys. Rev. B, 90, 125143, (2014)
[2] Asch, J; Bourget, O; Joye, A, Spectral stability of unitary network models, Rev. Math. Phys., 27, 1530004, (2015) · Zbl 1326.81035
[3] Asch, J., Bourget, O., Joye, A.: Chirality induced interface currents in the Chalker Coddington model. arXiv:1708.02120 · Zbl 1210.82033
[4] 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) · Zbl 0985.81137
[5] 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) · Zbl 1331.82065
[6] Chalker, JT; Coddington, PD, Percolation, quantum tunnelling and the integer Hall effect, J. Phys. C Solid State, 21, 2665, (1988)
[7] Delplace, P; Fruchart, M; Tauber, C, Phase rotation symmetry and the topology of oriented scattering networks, Phys. Rev. B, 95, 205413, (2017)
[8] Elbau, P; Graf, GM, Equality of bulk and edge Hall conductance revisited, Commun. Math. Phys., 229, 415-432, (2002) · Zbl 1001.81091
[9] Fruchart, M, Complex classes of periodically driven topological lattice systems, Phys. Rev. B, 93, 115429, (2016)
[10] Graf, G.M., Tauber, C.: Bulk-Edge correspondence for two-dimensional Floquet topological insulators. arXiv:1707.09212 · Zbl 1392.82008
[11] Ho, C-M; Chalker, JT, Models for the integer quantum Hall effect: the network model, the Dirac equation, and a tight-binding Hamiltonian, Phys. Rev. B, 54, 8708, (1996)
[12] Nathan, F; Rudner, MS, Topological singularities and the general classification of Floquet-Bloch systems, New J Phys., 17, 125014, (2015)
[13] 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) · Zbl 1037.81106
[14] Kitagawa, T, Topological phenomena in quantum walks: elementary introduction to the physics of topological phases, Quantum Inf. Process, 11, 1107-1148, (2012) · Zbl 1252.82088
[15] Pasek, M; Chong, YD, Network models of photonic Floquet topological insulators, Phys. Rev. B, 89, 075113, (2014)
[16] Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From \(K\)-theory to Physics. Springer International Publishing, Szwitzerland (2016) · Zbl 1342.82002
[17] Rordam, M., Larsen, F., Laustsen, N.: An Introduction to K-theory for C \^{∗}-Algebras. Cambridge University Press, Cambridge (2000) · Zbl 0967.19001
[18] Rudner, MS; Lindner, NH; Berg, E; Levin, M, Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X, 3, 031005, (2013)
[19] Ryu, S; Schnyder, AP; Furusaki, A; Ludwig, AWW, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys., 12, 065010, (2010)
[20] Schulz-Baldes, H; Kellendonk, J; Richter, T, Similtaneous quantization of edge and bulk Hall conductivity, J. Phys. A, 33, l27-l32, (2000) · Zbl 0985.81137
[21] Wegge-Olsen, N.E.: K-Theory and C \^{∗}-Algebras. Oxford University Press, Oxford (1993) · Zbl 0780.46038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.