We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.
We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.
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.
We study two-terminal transport through two-dimensional periodically driven systems in which all bulk Floquet eigenstates are localized by disorder. We focus on the anomalous Floquet-Anderson insulator (AFAI) phase, a topologically nontrivial phase within this class, which hosts topologically protected chiral edge modes coexisting with its fully localized bulk. We show that the unique properties of the AFAI yield remarkable far-from-equilibrium transport signatures: for a large bias between leads, a quantized amount of charge is transported through the system each driving period. Upon increasing the bias, the chiral Floquet edge mode connecting source to drain becomes fully occupied and the current rapidly approaches its quantized value.
We study localization and charge dynamics in a monochromatically driven one-dimensional Anderson insulator focusing on the low-frequency, strong-driving regime. We study this problem using a mapping of the Floquet Hamiltonian to a hopping problem with correlated disorder in one higher harmonic-space dimension. We show that (i) resonances in this model correspond to adiabatic Landau-Zener (LZ) transitions that occur due to level crossings between lattice sites over the course of dynamics; (ii) the proliferation of these resonances leads to dynamics that appear diffusive over a single drive cycle, but the system always remains localized; (iii) actual charge transport occurs over many drive cycles due to slow dephasing between these LZ orbits and is logarithmic in time, with a crucial role being played by far-off Mott-like resonances; and (iv) applying a spatially varying random phase to the drive tends to decrease localization, suggestive of weak-localization physics. We derive the conditions for the strong-driving regime, determining the parametric dependencies of the size of Floquet eigenstates, and time scales associated with the dynamics, and corroborate the findings using both numerical scaling collapses and analytical arguments.
Periodically driven systems often exhibit behavior distinct from static systems. In single-particle, static systems, any amount of disorder generically localizes all eigenstates in one dimension. In contrast, we show that in topologically nontrivial, single-particle Floquet loop drives with chiral symmetry in one dimension, a localization-delocalization transition occurs as the time t is varied within the driving period ($0\ensuremath{\le}t\ensuremath{\le}{T}_{\text{drive}}$). We find that the time-dependent localization length Lloc(t) diverges with a universal exponent as t approaches the midpoint of the drive: ${L}_{\mathrm{loc}}(t)\ensuremath{\sim}{(t\ensuremath{-}{T}_{\text{drive}}/2)}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=2$. We provide analytical and numerical evidence for the universality of this exponent within the AIII symmetry class.
Periodic driving provides a powerful route for controlling localization and engineering effective Hamiltonians in quantum systems. A localized one-dimensional Aubry-André chain can be driven into a fully delocalized phase when the entire system is driven below a critical frequency. Here, we ask whether such delocalization can be achieved by driving only a part of the system. We address this question using a two-leg ladder comprising an Aubry-André chain coupled to a periodically driven clean tight-binding chain. Exact numerical calculations show that, starting from a localized static ladder, subsystem driving below a critical frequency cannot achieve complete delocalization. Instead, a mixed phase emerges, with localized and delocalized states coexisting and the fraction of delocalized states varying nonmonotonically with the driving frequency. This behavior can be understood using local effective Hamiltonians. Although these effective Hamiltonians are derived analytically in the high-frequency regime, they can nevertheless predict the localization transition occurring in the moderate-frequency regime. In fact, by quantifying the emergent nonlocality of the Floquet Hamiltonian, we show that these effective Hamiltonians provide a valid description up to a crossover frequency that is lower than the critical frequency associated with the localization transition. Our results demonstrate an alternative route to Floquet engineering, in which the transport properties of a quantum lattice system can be modified by driving only a subsystem.
The anomalous Floquet Anderson insulator (AFAI) has been theoretically predicted in stepwise periodically driven models, but its stability under more general driving protocols has not been determined. We show that adding disorder to the anomalous Floquet topological insulator realized with a continuous driving protocol in the experiment by Wintersperger et al. [Nat. Phys. 16, 1058 (2020)], supports an AFAI phase, where, for a range of disorder strengths, all the time averaged bulk states become localized, while the pumped charge in a Laughlin pump setup remains quantized.