×

Eigenvalue crossings in Floquet topological systems. (English) Zbl 1434.35138

Summary: The topology of electrons on a lattice subject to a periodic driving is captured by the three-dimensional winding number of the propagator that describes time evolution within a cycle. This index captures the homotopy class of such a unitary map. In this paper, we provide an interpretation of this winding number in terms of local data associated with the eigenvalue crossings of such a map over a three-dimensional manifold, based on an idea from F. Nathan and M. S. Rudner [“Topological singularities and the general classification of Floquet-Bloch systems”, New J. Phys. 17, No. 12, Article ID 125014, 23 p. (2015; doi:10.1088/1367-2630/17/12/125014)]. We show that, up to homotopy, the crossings are a finite set of points and non-degenerate. Each crossing carries a local Chern number, and the sum of these local indices coincides with the winding number. We then extend this result to fully degenerate crossings and extended submanifolds to connect with models from the physics literature. We finally classify up to homotopy the Floquet unitary maps, defined on manifolds with boundary, using the previous local indices. The results rely on a filtration of the special unitary group as well as the local data of the basic gerbe over it.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
55M25 Degree, winding number
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnol’D, Vi, Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect, Selecta Mathematica, 1, 1, 1-19 (1995) · Zbl 0841.58008 · doi:10.1007/BF01614072
[2] Atiyah, Mf, On the K-theory of compact Lie groups, Topology, 4, 1, 95-99 (1965) · Zbl 0136.21001 · doi:10.1016/0040-9383(65)90051-0
[3] Bunk, S., Szabo, R.J.: Topological insulators and the Kane-Mele invariant: obstruction and localisation theory. arXiv preprint arXiv:1712.02991 (2017) · Zbl 1455.81028
[4] Brylinski, J-L, Loop Spaces, Characteristic Classes and Geometric Quantization (1993), MA: Birkhäuser Boston Inc, MA · Zbl 0823.55002
[5] Carpentier, D.; Delplace, P.; Fruchart, M.; Gawędzki, 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 · doi:10.1016/j.nuclphysb.2015.05.009
[6] De Nittis, G.; Gomi, K., Chiral vector bundles, Mathematische Zeitschrift, 290, 3-4, 775-830 (2018) · Zbl 1411.57044 · doi:10.1007/s00209-018-2041-1
[7] Duistermaat, Jj; Kolk, Jac, Lie groups Universitext (2000), Berlin: Springer, Berlin · Zbl 0955.22001
[8] Fuchs, J.; Schweigert, C., Symmetries, Lie algebras and Representations: A Graduate Course for Physicists (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 0923.17001
[9] Gawędzki, K.; Reis, N., WZW branes and gerbes, Rev. Math. Phys., 14, 12, 1281-1334 (2002) · Zbl 1033.81067 · doi:10.1142/S0129055X02001557
[10] Gawędzki, K.: Bundle gerbes for topological insulators. arXiv preprint arXiv:1512.01028 (2015) · Zbl 1397.53044
[11] Gawędzki, K., Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators, J. Geom. Phys., 120, 169-191 (2017) · Zbl 1430.53027 · doi:10.1016/j.geomphys.2017.05.017
[12] Golterman, Mf; Jansen, K.; Kaplan, Db, Chern-Simons currents and chiral fermions on the lattice, Phys. Lett. B, 301, 2-3, 219-223 (1993) · doi:10.1016/0370-2693(93)90692-B
[13] Graf, Gm; Tauber, C., Bulk-edge correspondence for two-dimensional Floquet topological insulators, Annales Henri Poincaré, 19, 3, 709-741 (2018) · Zbl 1392.82008 · doi:10.1007/s00023-018-0657-7
[14] Guillemin, V.; Pollack, A., Differential Topology (2010), Providence: American Mathematical Society, Providence · Zbl 1420.57001
[15] Hitchin, N.: Lectures on special Lagrangian submanifolds. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds Studies of Advanced Mathematical, vol. 23, pp. 151-182. American Mathematical Society, Providence (2001) · Zbl 1079.14522
[16] Höckendorf, B.; Alvermann, A.; Fehske, H., Efficient computation of the W 3 topological invariant and application to Floquet-Bloch systems, J. Phys. A Math. Theor., 50, 29, 295301 (2017) · Zbl 1372.82050 · doi:10.1088/1751-8121/aa7591
[17] Kohmoto, M., Topological invariant and the quantization of the Hall conductance, Ann. Phys., 160, 2, 343-354 (1985) · doi:10.1016/0003-4916(85)90148-4
[18] Loring, Ta; Schulz-Baldes, H., Finite volume calculation of K-theory invariants, N. Y. J. Math., 23, 1111-1140 (2017) · Zbl 1384.46048
[19] Meinrenken, E., The basic gerbe over a compact simple Lie group, Enseign. Math. (2), 49, 3-4, 307-333 (2003) · Zbl 1061.53034
[20] Milnor, J., Morse Theory (AM-51) (2016), Princeton: Princeton University Press, Princeton
[21] Monaco, D.; Tauber, C., Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess-Zumino, and Fu-Kane-Mele, Lett. Math. Phys., 107, 7, 1315-1343 (2017) · Zbl 1370.35093 · doi:10.1007/s11005-017-0946-y
[22] Mondragon-Shem, I.; Hughes, Tl; Song, J.; Prodan, E., Topological criticality in the chiral-symmetric AIII class at strong disorder, Phys. Rev. Lett., 113, 4, 046802 (2014) · doi:10.1103/PhysRevLett.113.046802
[23] Murray, Mk, Bundle gerbes, J. Lond. Math. Soc., 54, 2, 403-416 (1996) · Zbl 0867.55019 · doi:10.1112/jlms/54.2.403
[24] Nathan, F.; Rudner, Ms, Topological singularities and the general classification of Floquet-Bloch systems, New J. Phys., 17, 12, 125014 (2015) · doi:10.1088/1367-2630/17/12/125014
[25] Prodan, E.; Schulz-Baldes, H., Bulk and Boundary Invariants for Complex Topological Insulators. From K-Theory to Physics Mathematica Physics Studies (2016), Berlin: Springer, Berlin · Zbl 1342.82002
[26] 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, 3, 031005 (2013)
[27] Sadel, C.; Schulz-Baldes, H., Topological boundary invariants for Floquet systems and quantum walks, Math. Phys. Anal. Geom., 20, 4, 22 (2017) · Zbl 1413.46061 · doi:10.1007/s11040-017-9253-1
[28] Schreiber, U.; Schweigert, C.; Waldorf, K., Unoriented WZW models and holonomy of bundle gerbes, Commun. Math. Phys., 274, 1, 31-64 (2007) · Zbl 1148.53057 · doi:10.1007/s00220-007-0271-x
[29] Shapiro, J.; Tauber, C., Strongly disordered Floquet topological systems, Annales Henri Poincaré, 20, 6, 1837-1875 (2019) · Zbl 1482.82051 · doi:10.1007/s00023-019-00794-3
[30] Tauber, C., Effective vacua for Floquet topological phases: a numerical perspective on the switch-function formalism, Phys. Rev. B, 97, 19, 195312 (2018) · doi:10.1103/PhysRevB.97.195312
[31] Ünal, Fn; Seradjeh, B.; Eckardt, A., How to directly measure Floquet topological invariants in optical lattices, Phys. Rev. Lett., 122, 25, 253601 (2019) · doi:10.1103/PhysRevLett.122.253601
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.