Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems

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Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems

Mark S. Rudner, Netanel H. Lindner, Erez Berg, and Michael Levin

Phys. Rev. X 3, 031005 – Published 23 July 2013

Abstract

Recently, several authors have investigated topological phenomena in periodically driven systems of noninteracting particles. These phenomena are identified through analogies between the Floquet spectra of driven systems and the band structures of static Hamiltonians. Intriguingly, these works have revealed phenomena that cannot be characterized by analogy to the topological classification framework for static systems. In particular, in driven systems in two dimensions (2D), robust chiral edge states can appear even though the Chern numbers of all the bulk Floquet bands are zero. Here, we elucidate the crucial distinctions between static and driven 2D systems, and construct a new topological invariant that yields the correct edge-state structure in the driven case. We provide formulations in both the time and frequency domains, which afford additional insight into the origins of the “anomalous” spectra that arise in driven systems. Possibilities for realizing these phenomena in solid-state and cold-atomic systems are discussed.

Received 14 January 2013

DOI:https://doi.org/10.1103/PhysRevX.3.031005

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Mark S. Rudner^1,2,3, Netanel H. Lindner^3,4, Erez Berg^2,5, and Michael Levin⁶

¹The Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
³Institute of Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
⁴Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
⁵Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
⁶Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

Popular Summary

When driven by a strong time-varying force, a physical system can exhibit remarkable phenomena that defy intuition based on the system’s behavior in the absence of driving. A classical example is the inverted pendulum with a rigid support: Although the upward-pointing state (with the pendulum balanced on its support) is ordinarily unstable, this configuration can be stabilized if the support is vigorously shaken up and down. When a quantum-mechanical system is subjected to periodic driving, interesting new behaviors are possible as well. In particular, particle transport along the surfaces of insulators can be drastically modified by the application of strong electromagnetic radiation. Robust edge transport is a well-known phenomenon occurring in a special class of (nondriven) materials called topological insulators. In this theoretical paper, we identify the fundamental differences between the driven and nondriven cases, and explain the mathematical structure of quantum states that governs the appearance of conducting edge modes in driven systems.

For a static (i.e., nondriven) topological insulator, a set of “Chern numbers” characterizes the nontrivial topology of the electronic band structure. This set of numbers allows a complete and unambiguous determination of the presence or absence of conducting edge states based on the bulk properties of the insulator. Focusing on models of two-dimensional electronic systems of noninteracting particles, we have demonstrated that when such a system is periodically driven, the Chern numbers no longer uniquely determine the appearance of conducting edge states. Indeed, new types of edge modes appear even when the Chern numbers of all bands are zero. Resolving this mysterious behavior, we have identified a new topological invariant, a winding number, which acts as the analogue of the Chern number for driven topological insulators.

Experimental investigations of the new physics that we describe could be feasible in both solid-state and cold-atom systems. Many open theoretical questions about the role of particle-particle interactions and additional symmetries remain to be explored.

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Vol. 3, Iss. 3 — July - September 2013

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Figure 1
Topologically protected Floquet edge modes in a two-band system with trivial Chern indices $C = 0$ . This situation is made possible due to the periodicity of the quasienergy $ε$ : In addition to the edge modes that cross the gap near $ε = 0$ , a second branch of edge modes crosses through $ε = π / T$ , thus connecting the top of the upper Floquet band to the bottom of the lower Floquet band and closing the quasienergy cycle. The Chern numbers correctly yield the differences between the numbers of chiral modes above and below every band but cannot be used to uniquely determine the edge-state spectrum of a periodically driven system.Reuse & Permissions
Figure 2
Two-dimensional tight-binding model exhibiting anomalous Floquet edge modes. (a) Driving protocol. Hopping amplitudes are varied in a spatially homogeneous but chiral, time-periodic way. A sublattice potential $δ_{A B}$ differentiating the two types of sites in the bipartite lattice (filled and open circles) is applied throughout. During four of the five equal-length phases of the cycle, hopping along one of the four distinct bond types is allowed, with amplitude $J$ (bold lines). During the fifth phase, all hopping amplitudes are zero, while the sublattice potential remains constant. (b) In the simple case $J T = 5 π / 2$ , $δ_{A B} = 0$ , particles in the bulk move in closed trajectories encircling plaquettes, and the Floquet operator is the identity. Chiral edge modes propagate along the boundaries. (c) Floquet spectrum for the case described in (b).Reuse & Permissions
Figure 3
Phases of the square lattice model in Sec. 3. (a)–(c) Example Floquet spectra for each of the three phases, calculated in a strip geometry (unit cell $2 a$ ), with sublattice potential $δ_{A B} = 0.5 π / T$ and hopping amplitudes (a) $J = 0.5 π / T$ , (b) $J = 1.5 π / T$ , and (c) $J = 2.5 π / T$ . The winding numbers $W_{0}$ and $W_{π}$ correctly yield the numbers of edge states in the two gaps. The asymmetry in $k_{∥}$ is due to the breaking of inversion symmetry by the sublattice potential $δ_{A B}$ . (d) Phase diagram indicating the winding numbers $W_{0}$ and $W_{π}$ , calculated in the gaps at quasienergies 0 and $π / T$ , and Chern number $C$ of the upper band. Note the existence of two topologically distinct phases with $C = 0$ .Reuse & Permissions
Figure 4
Floquet Hamiltonian and level diagram in the repeated zone scheme. (a) When the driving $Δ (t)$ consists of a single harmonic with angular frequency $ω$ , the Floquet Hamiltonian $H (k)$ [see Eq. (18)] assumes a block-tridiagonal form. Each block is an $N \times N$ matrix, where $N$ is the number of bands in the unperturbed Hamiltonian $H_{0}$ . The off-diagonal blocks labeled by $Δ$ and $Δ^{†}$ describe transitions accompanied by the absorption and emission of a driving field photon, respectively. (b) Schematic Floquet level diagram. The $N$ -level spectrum of $H_{0} (k)$ , which spans an energy range $Λ$ , is copied and rigidly shifted by $m ω$ for each value of $m$ . The harmonic driving $Δ$ induces transitions between neighboring copies with $m$ values differing by 1. Analogous to Wannier-Stark states, the Floquet eigenstates in the first quasienergy Brillouin zone (gray shaded region) are localized in $m$ , with amplitudes highly suppressed for $| m | ≫ Λ / ω$ .Reuse & Permissions
Figure 5
Floquet bands for a two-band model in the truncated repeated zone scheme. Representative one-dimensional cuts through the two-dimensional crystal-momentum Brillouin zone are shown. (a) For $Δ = 0$ , the unperturbed bands of $H_{0}$ are each copied and shifted up and down one time; $m = - 1, 0, 1$ . The bands have Chern numbers $C = \pm C_{0}$ . The crossings between bands with $m$ values differing by 1 correspond to points in the Brillouin zone where the driving field resonantly couples the two bands. (b) For $Δ \neq 0$ , a generic driving field opens avoided crossings at the resonance points. After coupling, the Chern numbers of the crossed bands are changed to $C = \pm C_{F}$ . In the truncated scheme, the appearance of Floquet edge states at a particular quasienergy in the first quasienergy Brillouin zone (gray regions) can be predicted by adding up the Chern numbers of all bands below it.Reuse & Permissions
Figure 6
Anomalous edge states in a weakly driven two-band system, with Hamiltonian $H_{0}$ given by Eqs. (20, 21). The Chern numbers of the unperturbed bands are $C_{0} = \pm 1$ . (a) Spectrum of $H_{0}$ with periodic boundary conditions, showing the resonance curve (purple line) in the valence and conduction bands. The color scheme indicates $⟨ σ_{z} ⟩$ for the corresponding eigenstates in the two bands. (b) Spectrum of $H_{0}$ for a cylindrical geometry, with periodic boundary conditions in the $x$ direction and open boundary conditions in the $y$ direction. (c) Spectrum of the truncated Floquet Hamiltonian [Eq. (18)] in the cylindrical geometry. The isolated bands at the top and bottom of the figure do not participate in any resonances and are nearly identical to the original bands of $H_{0}$ . The Floquet bands have zero Chern numbers, while edge states traverse both gaps, centered around $ε = 0$ and $ε = π / T$ . The parameters used are $J / μ = b / μ = 1.5$ , $a / μ = 4$ , $Δ_{0} / μ = 1$ , and $Δ_{0} / ω = 0.07$ .Reuse & Permissions

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