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| Fig. 1: a) Schematic of the RLBL model where hopping is nonzero in each of the T/5 timesteps of the period for any colored bonds on the square lattice. b) Numerical quasienergy spectrum of the RLBL model solved in a strip geometry for small hopping; both bands have Chern number zero and no edge states are observed. c) Numerical quasienergy spectrum of the RLBL model solved in a strip geometry for large hopping; both bands have Chern number zero but chiral edge states are observed in both gaps. d) Band structure of two-band model with Chern numbers ± solved numerically in a strip geometry. e) Floquet spectrum of model from d) driven at frequency ω and truncated at ± ω. Edge states are observed crossing in each gap, even between zero Chern bands. Units of quasienergy axis are in terms of ω. (Source: C. Foutty, after Rudner et al. [1]) |
A detailed theoretical understanding of topological phases of matter is a key development in condensed matter physics over the last 40 years. As periodically driven, Floquet systems gain widespread interest, a logical question is how these topological phases extend to quasiperiodic Floquet spectra. We will review a paper by Rudner et al. that provides a generalization of the bulk-boundary correspondence to Floquet systems and provide numerical simulations of these sorts of edge states in multiple kinds of models. [1] After developing this single-particle picture, we will discuss how these topological Floquet phases might extend to many-body/physical systems with the addition of interactions.
First, it's necessary to review the topological classification for static systems that gives rise to chiral edge states. Single-particle theory gives rise to band theory and corresponding energy bands. If these bands are separated from one another by an energy gap, then each band can be identified with an integer Chern number, which quantifies (in units of 2π) the integral of the Berry curvature over the entire Brillouin zone. The Chern number, which is defined directly from the bulk band structure, is closely tied to the existence of edge states through the "bulk-boundary correspondence": the Chern number of each energy band is equivalent to the difference between the number of chiral edge modes above and below the band. If we have an entire band structure of isolated bands, then, the net number of chiral edge modes is given by the sum over the Chern number of all occupied bands. Importantly, the existence and number of edge states in the static system is completely defined by the Chern numbers of each energy band; nonzero Chern numbers are required to observe chiral edge modes.
Floquet systems, for which the Hamiltonian is periodic with period T, are characteristically different from time-independent systems. In an analogy to Bloch's theorem in systems with discrete spatial translation symmetry, general eigenstates of a Floquet Hamiltonian can be expressed as
The quasienergy ε can form quasienergy bands in momentum space, but it is only defined up to integer multiples of 2π/T. The entire Floquet spectrum can then be defined on a periodic quasienergy Brillouin zone -π/T < ε < π/T. This periodicity leads to very different behavior than in static case, as there are no isolated bands at the top and bottom of the spectrum.
Just as in the static case, one can evaluate the Chern numbers on each quasienergy band of the Floquet spectrums at a particular instance in time. However, this does not completely specify the edge state behavior of the Floquet spectra, due to the periodicity of the quasienergy Brillouin zone. Consider a case with only two Floquet bands. If there is a single chiral edge state connecting the two bands through the center of the quasienergy zone and another chiral edge state connecting the two around the periodic boundary, then both bands will have zero Chern number. But they still display "anomalous" edge states in each energy gap that could not have been predicted by the Chern numbers alone.
To expand on this, Rudner et al. define a Floquet tight-binding model, schematically represented in Fig. 1a. [1] The hopping on a bipartite lattice is piecewise in time, with specific hoppings set to a constant value for each of the first four T/5 timesteps and then a resting period on the fifth T/5 time step. For a specific choice of parameters, including setting the sublattice energy difference to zero, a particle will hop with probability 1 between neighboring sites on each unit of the cycle. So in an infinite geometry, the time evolution over the Floquet period will just be the identity, as every particle will hop around a square plaquette with direction controlled by the chirality of the drive. However, with the introduction of a finite edge, particles can no longer hop off of the edge, and each edge must host a chiral edge state propagating in opposite directions. For different choices of sublattice energy difference and hopping parameters, different phases can be tuned to (see Fig. 1b-1c), and we can solve these numerically in a strip geometry by taking finite, open boundary conditions in one direction and periodic boundary conditions in the other. In Fig. 1b, the bands are completely separated and have Chern number zero, displaying no edge states. In Fig. 1c, however, both bands have Chern number zero but singular chiral edge states, localized on each edge in the cylindrical geometry, crossing both gaps. The existence of these edge states even while the Chern numbers are zero motivates a new form of topological invariant that classifies the edge states based on the topological properties of the bulk for Floquet systems.
In order to construct an invariant, Rudner et al. consider two possible cases - the first is that the time evolution operator given by the integration of the Schrödinger equation satisfies U(T) = 1, as in the first case of the square lattice model discussed above, and the other case is that the time evolution operator doesn't satisfy this condition. For the first case, the time evolution is periodic in momentum coordinates and time, so that it forms a map from a direct product of circles to a unitary group. This sort of map can be classifed by an integer winding number given by
where this winding number can be shown to be equivalent to the number of chiral edge states. In the more complicated case where U(T) ≠ 1, we can interpolate between U and a Uε that satisfies the condition Uε(T) = 1, where the interpolation depends on the energy of the gap examined, and then the winding number of Uε gives the number of chiral edge states at quasienergy ε. Though the definition and derivation is quite technical, the result is a classification that integrates over the time evolution across the entire period of the Floquet drive, which is thus able to include the inherently dynamical chiral edge states that cannot be predicted by Chern numbers alone.
One case of interest that has a simpler interpretation is that of a harmonic drive at a particular frequency, which is relevant to many physically realizable cases. In general, the Floquet Hamiltonian can be decomposed in frequency space and then diagonalized effectively in an "extended zone" scheme if truncated at a finite momentum, rather than with periodic boundaries. If the drive is at a single frequency, the evaluation in frequency-space is particularly straightforward. For small magnitude of drives, this approach results in multiple copies of the original, static Hamiltonian shifted up and down by energy determined by the frequency of the drive. Higher orders in perturbation theory lead to gaps opening at crossings of the bands. Once this band structure is truncated, it can be treated as if it were a static band structure and edge modes can be counted by summing the Chern numbers of all bands below a certain gap. Anomalous edge states can still arise between zero Chern number Floquet bands if the original band structure had nontrivial topology; this is shown in Fig 1d-1e. Fig. 1d shows the band structure of a particular 2-band model chosen so that the Chern numbers of the bands are ± 1, solved for numerically in a strip geometry. Fig. 1e shows the Floquet band structure truncated at ±ω, so that all of the internal Floquet bands - not on the top or bottom - have zero Chern number, yet edge states are still observed crossing between each of them.
While the single-particle picture in a closed system allows for a detailed theoretical understanding of topological edge modes in Floquet systems, it's worth discussing how these might change in a system with interactions. A recent review from Harper et al. discusses some of these implications. [2] In principle, a closed, interacting Floquet system should heat to infinite temperature. Without energy conservation, entropy is always maximized by the infinite temperature state. In this case, any interesting topological properties would be essentially unmeasurable under the thermalization to a completely trivial state. However, there are some possibilities that avoid devolution into the infinite-energy state: many-body localization (MBL), prethermalization, and cooling. Sufficient interactions in a many-body Floquet state should lead to an MBL phase somewhat analogously to the static case. Making this definition more precise requires re-expressing the l-bit MBL formalism in terms of time evolution operators U(T) and extending the eigenstate thermalization hypothesis (ETH) to Floquet eigenstates. While this Floquet-MBL phase is expected, via preliminary numerical and experimental studies, to exist, it's unclear to what extent delocalized topological edge states might be compatible with Floquet-MBL. Prethermalization is the case that heating to the steady state takes parametrically large time compared to relevant energy scales. Generally, heating rates will be small if the drive frequency is much larger than the relevant energy scales of the problem, such as the single-particle bandwidth. Finally, in an open system, one expects the system to thermalize to some steady state that's not necessarily infinite temperature, due to coupling to some sort of external bath. Because the system is inherently out of equilibrium, the precise steady state will depend sensitively on the microscopics of the bath and the system-bath coupling, rather than on a small number of macroscopic parameters (e.g. temperature, chemical potential) that you might have in the static case. Most early experiments in to topological Floquet systems have focused on ultrafast measurement to avoid the long-time dynamics of the Floquet systems, though they remain of physical interest.
© Ben Foutty. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] M. S. Rudner et al., "Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems," Phys. Rev. X 3, 031005 (2013).
[2] F. Harper et al., "Topology and Broken Symmetry in Floquet Systems," Annu. Rev. Condens. Matter Phys. 11, 345 (2020)