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.
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.
In thermal equilibrium, the interplay of symmetry and topology leads to a handful of topological insulating phases of matter with robust properties governed by topological invariants. This work investigates the nonequilibrium dynamics of three-dimensional systems of bosons, spins, or qubits, subject to time-periodic driving, and theoretically discovers an infinite family of new dynamical topological insulating phases that would not be possible in thermal equilibrium settings. These phases, called ``Floquet topological paramagnets'', display anomalous surface dynamics, including magnetic domain walls that transport unidirectionally precisely quantized packets of quantum information per driving period, and new types of dynamically enriched topological orders.
We consider topological phases in periodically driven (Floquet) systems exhibiting many-body localization, protected by a symmetry G. 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 accounts for the discrete time-translation symmetry. Thus, for example, the bosonic phases in d 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.
Phases of matter are traditionally seen as families of static systems exhibiting the same long-distance and low-energy correlations. In this work, the authors propose and classify a new family of phases of matter. They are novel insofar as they are intrinsically driven and out of equilibrium --- they can only be realized in systems with time-dependent Hamiltonians. The phases we consider arise in 1D periodically driven systems, in the presence of strong disorder and interactions, and are similar to but qualitatively distinct from the symmetry-protected topological phases now well known in the equilibrium setting. In a companion paper, the authors examine similar intrinsically driven families of states with long-range order, and an order parameter that oscillates at a frequency that is robustly an integer multiple of the underlying drive frequency.
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.
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 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 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.
This thesis presents studies of the role of disorder in non-equilibrium quantum systems. The quantum states relevant to dynamics in these systems are very different from the ground state of the Hamiltonian. Two distinct systems are studied, (i) periodically driven Hamiltonians in two dimensions, and (ii) electrons in a one-dimensional lattice with power-law decaying hopping amplitudes. In the first system, the novel phases that are induced from the interplay of periodic driving, topology and disorder are studied. In the second system, the Anderson transition in all the eigenstates of the Hamiltonian are studied, as a function of the power-law exponent of the hopping amplitude. In periodically driven systems the study focuses on the effect of disorder in the nature of the topology of the steady states. First, we investigate the robustness to disorder of Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are generated by resonantly driving a transition between the valence and conduction band. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a gap at the resonant quasienergy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator. Interestingly, the effects of disorder are not necessarily adverse, disorder can also induce a transition from a trivial to a topological system, thereby establishing a Floquet Topological Anderson Insulator (FTAI). Such a state would be a dynamical realization of the topological Anderson insulator. We identify the conditions on the driving field necessary for observing such a transition. We realize such a disorder induced topological Floquet spectrum in the driven honeycomb lattice and quantum well models. Finally, we show that two-dimensional periodically driven quantum systems with spatial disorder admit a unique topological phase, which we call the anomalous Floquet-Anderson insulator (AFAI). The AFAI is characterized by a quasienergy spectrum featuring chiral edge modes coexisting with a fully localized bulk. Such a spectrum is impossible for a time-independent, local Hamiltonian. These unique characteristics of the AFAI give rise to a new topologically protected nonequilibrium transport phenomenon: quantized, yet nonadiabatic, charge pumping. We identify the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition. The thesis also present the study of disordered systems using Wegner's Flow equations. The Flow Equation Method was proposed as a technique for studying excited states in an interacting system in one dimension. We apply this method to a one-dimensional tight binding problem with power-law decaying hoppings. This model presents a transition as a function of the exponent of the decay. It is shown that the the entire phase diagram, i.e. the delocalized, critical and localized phases in these systems can be studied using this technique. Based on this technique, we develop a strong-bond renormalization group that procedure where we solve the Flow Equations iteratively. This renormalization group approach provides a new framework to study the transition in this system.