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arXiv:1602.08089 · Physical review. B./Physical review. B

Abelian Floquet symmetry-protected topological phases in one dimension

Rahul Roy, Fenner Harper

Published Sep 2, 2016• 121 citations
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Abstract

Time-dependent systems have recently been shown to support novel types of topological order that cannot be realized in static systems. In this paper we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes, which have signatures in the many-body quasienergy spectrum and which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realize nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.

DOI
10.1103/physrevb.94.125105

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Time-dependent systems have recently been shown to support novel types of topological order that cannot be realized in static systems. In this paper we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes, which have signatures in the many-body quasienergy spectrum and which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realize nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.
Time-dependent systems have recently been shown to support novel types of topological order that cannot be realised in static systems. In this paper, we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes in the many-body quasienergy spectrum, which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realise nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.
Periodically driven Floquet systems can host dynamical phases, including a range of exotic topological phases that have no static analogs. This work presents a homotopy approach to the study of driven systems, which treats unitary evolutions as paths in the space of unitary operators. By considering loop evolutions in this space, the authors obtain a topological classification, free of uncertainties and ambiguities about the long-time robustness of of behavior in specific physical models. Two classes of Floquet symmetry-protected topological phase are identified, characterized by unitary evolutions that act trivially in the bulk but nontrivially at the boundary of an open system. The first class is captured by an explicit group cohomology construction; it has the remarkable property of converting a trivial boundary state into a nontrivial symmetry-protected topological state. The second class exhibits anomalous counter propagating Hilbert-space transport at the boundary; it lies beyond the cohomology construction.
Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and $\ensuremath{\pi}$ modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized $\ensuremath{\pi}$ modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.
Floquet systems display rich phenomena, such as time crystals, with many-body localisation (MBL) protecting the phases from heating. While several types of Floquet phases have been classified, a unified picture of Floquet MBL is still emerging. Static phases have been fruitfully studied via "symmetry topological field theory" (SymTFT), wherein the universal features of GGG-symmetric systems are elucidated by placing them on the boundary of a topological order of one dimension higher. In this work, we provide a SymTFT approach to classifying GGG-symmetric Floquet MBL phases in 1D, for GGG a finite Abelian group with on-site unitary action. In the SymTFT, these 1D systems correspond to the boundaries of the quantum double associated to GGG, and the classification naturally arises from considering the Lagrangian subgroups and boundary excitations of the quantum double. The classification covers all known Floquet phases while uncovering others previously unexplored, along with bulk features of phases thought to have only boundary signatures. We refer to the latter phases as "dual" time crystals. For static phases, we show how anyons of the quantum double and (string) order parameters provide a natural and simple interpretation of known classification schemes. By extending our framework to the boundaries of twisted quantum doubles, we uncover a new time-crystalline phase with non-onsite symmetry, which cannot be obtained through local, symmetric Hamiltonian drives. We numerically demonstrate evidence for the absolute stability of this phase, and observe that for open boundary conditions it has greater stability to symmetric perturbations. We finally discuss perspectives on using programmable quantum devices to realise and probe the phases we discuss. Our results show that SymTFT provides a powerful approach to unifying phases and features of Floquet systems.
We find a class of Floquet topological phases exhibiting gap-dependent topological classifications in quantum systems with a dynamical space-time symmetry and an antisymmetry. This is in contrast to all existing Floquet topological phases protected by static symmetries, where the topological classification across all quasienergy gaps is characterized by the same Abelian group. We demonstrate this gap-dependent classification phenomenon using the frequency-domain formulation of the time-dependent Hamiltonian. Moreover, we provide an interpretation of the resulting Floquet topological phases using a frequency lattice with a decoration represented by color degrees of freedom on the lattice vertices. These colors correspond to the coefficient NNN of the group extension of the system symmetry group GGG along the frequency lattice, given by $N=Z\ifmmode\times\else\texttimes\fi{}{H}^{1}[A,M]$. The distinct topological classifications that arise at different energy gaps in its quasienergy spectrum are described by the torsion product of the cohomology group H2[G,N]{H}^{2}[G,N]H2[G,N] classifying the group extension.
While periodically driven phases offer a unique insight into nonequilibrium topology that is richer than its static counterpart, their experimental realization is often hindered by ubiquitous decoherence effects. Recently, we have proposed a decoherence-free approach of realizing these Floquet phases. The central insight is that the reflection matrix, being unitary for a bulk insulator, plays the role of a Floquet time-evolution operator. We have shown that reflection processes off the boundaries of systems supporting higher-order topological phases (HOTPs) simulate nontrivial Floquet phases. So far, this method was shown to work for one-dimensional Floquet topological phases protected by local symmetries. Here, we extend the range of applicability by studying reflection off three-dimensional HOTPs with corner and hinge modes. We show that the reflection processes can simulate both first-order and second-order Floquet phases, protected by a combination of local and spatial symmetries. For every phase, we discuss appropriate topological invariants calculated with the nested scattering matrix method.