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arXiv:1703.02553 · Physical review. B./Physical review. B

Floquet topological phases protected by time glide symmetry

Takahiro Morimoto, Hoi Chun Po, Ashvin Vishwanath

Published May 26, 2017• 105 citations
arXivHTMLPDF

Abstract

Nonequilibrium systems under periodic driving realize novel topological phases that cannot be achieved in equilibrium systems. One unique feature of periodically driven systems is that they can host a purely dynamical symmetry that involves time translation. This work explores a new class of Floquet topological phases protected by one realization of such dynamical symmetry, i.e., ``time glide symmetry'', which is defined by a combination of reflection and time translation. Lattice models with time glide symmetric driving that are introduced show stable gapless surface states along with nontrivial topological numbers defined with time glide symmetry. In addition, a general classification theory of time glide symmetric topological phases is obtained by using a Clifford algebra approach.

DOI
10.1103/physrevb.95.195155

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Nonequilibrium systems under periodic driving realize novel topological phases that cannot be achieved in equilibrium systems. One unique feature of periodically driven systems is that they can host a purely dynamical symmetry that involves time translation. This work explores a new class of Floquet topological phases protected by one realization of such dynamical symmetry, i.e., ``time glide symmetry'', which is defined by a combination of reflection and time translation. Lattice models with time glide symmetric driving that are introduced show stable gapless surface states along with nontrivial topological numbers defined with time glide symmetry. In addition, a general classification theory of time glide symmetric topological phases is obtained by using a Clifford algebra approach.
Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.
Floquet engineering is one of the most vigorous fields in periodically driven (Floquet) systems, with which we can control phases of matter usually by high-frequency drives. In this paper, with Floquet engineering by a combination of high-frequency drives and resonant drives, we propose a way to realize nontrivial topological phases protected by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry only in the presence of a Z2{\mathbb{Z}}_{2}Z2​ symmetry, using a robust emergent Z2{\mathbb{Z}}_{2}Z2​ symmetry induced by the resonant drives. Moreover, the symmetry protected topological (SPT) phases are switchable between nontrivial and trivial phases only by the direction of a static transverse field, and even perturbations on the resonant drive can be utilized to realize richer SPT phases. We also discuss the real-time dynamics of the model, and find that which topological phases the system lies in can be distinguished by a period doubling of a nonlocal order parameter, as with discrete time crystals. A realization or a control of nontrivial SPT phases without the required symmetries by resonant drives, proposed in this paper, would shed a light on the observation of topological phenomena in nonequilibrium setups.
We consider topological phases in periodically driven (Floquet) systems exhibiting many-body localization, protected by a symmetry GGG. We argue for a general correspondence between such phases and topological phases of undriven systems protected by symmetry $\mathbb{Z}\ensuremath{\rtimes}G$ where the additional Z\mathbb{Z}Z accounts for the discrete time-translation symmetry. Thus, for example, the bosonic phases in ddd spatial dimensions without intrinsic topological order [symmetry-protected topological (SPT) phases] are classified by the cohomology group ${H}^{d+1}[\mathbb{Z}\ensuremath{\rtimes}G,\mathrm{U}(1)]$. For unitary symmetries, we interpret the additional resulting Floquet phases in terms of the lower-dimensional SPT phases that are pumped to the boundary during one time step. These results also imply the existence of novel symmetry-enriched topological (SET) orders protected solely by the periodicity of the drive.
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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.
Time-periodic driving enables new nonequilibrium quantum phases of matter with topologically protected and quantum coherent motion that would be forbidden in thermal equilibrium. This work uncovers two new classes of driven topological phases in driven two-dimensional spin systems: i) Floquet symmetry protected topological phases, which are driven and interacting analogs of topological insulators whose edge states are protected against disorder and localization; and ii) Floquet enriched topological orders, in which time-dependent driving ``breaks'' the spins into anyonic particles with fractional statistics and oscillating topological charge. The physics of these phases is explored by constructing toy models that can be explicitly solved despite the presence of strong interactions.