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arXiv:1908.04100 · Physical Review A

Floquet engineering of topological phases protected by emergent symmetries under resonant drives

Kaoru Mizuta, Kazuaki Takasan, Norio Kawakami

Published Nov 18, 2019• 0 citations
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Abstract

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 ${\mathbb{Z}}_{2}$ symmetry, using a robust emergent ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physreva.100.052109

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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 Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​ 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 propose and analyze two distinct routes toward realizing interacting symmetry-protected topological (SPT) phases via periodic driving. First, we demonstrate that a driven transverse-field Ising model can be used to engineer complex interactions which enable the emulation of an equilibrium SPT phase. This phase remains stable only within a parametric time scale controlled by the driving frequency, beyond which its topological features break down. To overcome this issue, we consider an alternate route based upon realizing an intrinsically Floquet SPT phase that does not have any equilibrium analog. In both cases, we show that disorder, leading to many-body localization, prevents runaway heating and enables the observation of coherent quantum dynamics at high energy densities. Furthermore, we clarify the distinction between the equilibrium and Floquet SPT phases by identifying a unique micromotion-based entanglement spectrum signature of the latter. Finally, we propose a unifying implementation in a one-dimensional chain of Rydberg-dressed atoms and show that protected edge modes are observable on realistic experimental time scales.
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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.
Periodic driving has a longstanding reputation for generating exotic phases of matter with no static counterparts. This work explores the interplay between periodic driving, interaction effects, and Z2{\mathbb{Z}}_{2}Z2​ symmetry that leads to the emergence of Floquet symmetry protected second-order topological phases in a simple but insightful two-dimensional spin-1/2 lattice. Through a combination of analytical and numerical treatments, we verify the formation of corner-localized 0 and $\ensuremath{\pi}$ modes, i.e., Z2{\mathbb{Z}}_{2}Z2​ symmetry broken operators that commute and anticommute, respectively, with the one-period time evolution operator, as well as establish the topological nature of these modes by demonstrating their presence over a wide range of parameter values and explicitly deriving their associated topological invariants under special conditions. Finally, we propose a means to detect the signature of such modes in experiments, and we discuss the effect of imperfections.
Symmetry-protected topological (SPT) phases of matter have been the focus of many recent theoretical investigations, but controlled mechanisms for engineering them have so far been elusive. In this work, we demonstrate that by driving interacting spin systems periodically in time and tuning the available parameters, one can realize lattice models for bosonic SPT phases in the limit where the driving frequency is large. We provide concrete examples of this construction in one and two dimensions, and discuss signatures of these phases in stroboscopic measurements of local observables.
We show how second-order Floquet engineering can be employed to realize systems in which many-body localization coexists with topological properties in a driven system. This allows one to implement and dynamically control a symmetry protected topologically ordered qubit even at high energies, overcoming the roadblock that the respective states cannot be prepared as ground states of nearest-neighbor Hamiltonians. Floquet engineering-the idea that a periodically driven nonequilibrium system can effectively emulate the physics of a different Hamiltonian-is exploited to approximate an effective three-body interaction among spins in one dimension, using time-dependent two-body interactions only. In the effective system, emulated topology and disorder coexist, which provides an intriguing insight into the interplay of many-body localization that defies our standard understanding of thermodynamics and into the topological phases of matter, which are of fundamental and technological importance. We demonstrate explicitly how combining Floquet engineering, topology, and many-body localization allows one to harvest the advantages (time-dependent control, topological protection, and reduction of heating, respectively) of each of these subfields while protecting them from their disadvantages (heating, static control parameters, and strong disorder).