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Bulletin of the American Physical Society

Stroboscopic Symmetry-Protected Topological Phases

Luiz H. Santos, Thomas Iadecola, Claudio Chamon

Published Mar 15, 2016• 0 citations
OpenAlex

Abstract

March, 2015)Symmetry-protected topological (SPT) phases of matter have been the focus of many recenttheoretical 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 andtuning the available parameters, one can realize lattice models for bosonic SPT phases in the limitwhere the driving frequency is large. We provide concrete examples of this construction in oneand two dimensions, and discuss signatures of these phases in stroboscopic measurements of localobservables.

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arXiv:1908.04100•Physical Review A•Nov 18, 2019•0 citations
March, 2015)Symmetry-protected topological (SPT) phases of matter have been the focus of many recenttheoretical 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 andtuning the available parameters, one can realize lattice models for bosonic SPT phases in the limitwhere the driving frequency is large. We provide concrete examples of this construction in oneand two dimensions, and discuss signatures of these phases in stroboscopic measurements of localobservables.
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.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
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.
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
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.
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.