The impact of weak disorder and its spatial correlation on the topology of a Floquet system is not well understood so far. In this study, we investigate a model closely related to a two-dimensional Floquet system that has been realized in experiments. In the absence of disorder, we determine the phase diagram and identify an exotic phase characterized by edge states with alternating chirality in adjacent gaps. When weak disorder is introduced, we examine the disorder-averaged Bott index and analyze why the anomalous Floquet topological insulator is favored by both uncorrelated and correlated disorder, with the latter having a stronger effect. For a system with a ring-shaped gap in the energy spectrum, the Born approximation fails to explain the topological phase transition, unlike for a system with a pointlike gap.
The impact of weak disorder and its spatial correlation on the topology of a Floquet system is not well understood so far. In this study, we investigate a model closely related to a two-dimensional Floquet system that has been realized in experiments. In the absence of disorder, we determine the phase diagram and identify an exotic phase characterized by edge states with alternating chirality in adjacent gaps. When weak disorder is introduced, we examine the disorder-averaged Bott index and analyze why the anomalous Floquet topological insulator is favored by both uncorrelated and correlated disorder, with the latter having a stronger effect. For a system with a ring-shaped gap in the energy spectrum, the Born approximation fails to explain the topological phase transition, unlike for a system with a pointlike gap.
Two-dimensional periodically driven topological insulators have been shown to exhibit numerous topological phases, including ones which have no static analog, such as anomalous Floquet topological phases. We study a two-dimensional model of spinless fermions on a honeycomb lattice with periodic driving. We show that this model exhibits a rich mixture of weak and strong topological phases, which we identify by computing their scattering matrix invariants. Furthermore, we do an in-depth analysis of these topological phases in the presence of spatial disorder and show the relative robustness of these phases against imperfections. Making use of this robustness against spatial disorder, we propose a filter which allows the passage of only edge states, and which can be realized using existing experimental techniques.
One of the most fascinating properties of topological phases of matter is their robustness to disorder and imperfections. Although several experimental techniques have been developed to probe the geometric properties of engineered topological Bloch bands with cold atoms, they almost exclusively rely on the translational invariance of the underlying lattice. This prevents direct studies of topology in the presence of disorder, further hindering an extension to disordered interacting topological phases. Here, we identify disorder-driven phase transitions between two distinct Floquet topological phases using the characteristic properties of topological edge modes with ultracold atoms in periodically-driven two-dimensional (2D) optical lattices. Our results constitute an important step towards studying the rich interplay between topology and disorder with cold atoms. Moreover, our measurements confirm that disorder indeed favors the anomalous Floquet topological regime over conventional Hall systems, indicating an enhanced robustness and paving the way towards observing exotic out-of-equilibrium phases such as the anomalous Floquet Anderson insulator.
We explore the physics of a Chern insulator subjected to a two-step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by the presence and chirality of edge modes in the two bulk gaps of the Floquet quasienergy spectrum, around 0 and $\ensuremath{\pi}$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically nontrivial time evolution. An extensive numerical study of the bulk topological invariants in the presence of quenched disorder reveals transitions induced by strong disorder (i) from the different topological to trivial insulator phases and (ii) from a trivial to a topological Anderson insulator phase at intermediate disorder strengths. Careful analysis of level statistics of the quasienergy spectrum indicates a ``levitation-annihilation'' mechanism near these transitions.
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.
Current research on the disordered topological quantum phases primarily focuses on the uncorrelated and short-range correlated disorder regime. These topological Anderson insulators are typically self-averaging. However, topological quantum systems with long-range correlated disorder have received limited attention due to the absence of tractable analytical methods. In fact, the long-range correlated disorder introduces more complex effects on topological quantum states. Here, we demonstrate that the long-range correlated disorder could induce anomalously statistical feature where the topological Anderson states become non-self-averaging, and the phase diagram is strongly dependent on individual disorder configurations. We term this statistical phase the non-self-averaging topological Anderson insulator. The non-self-averaging property is identified by the non-vanishing finite values of the relative variance of the Lyapunov exponent in the thermodynamic limit, alongside the non-Gaussian distributions of the Lyapunov exponent. Consequently, the topological properties of a single disordered sample deviate from the ensemble average, causing a breakdown of the central limit theorem. The non-self-averaging topological Anderson insulator provides insights into the interplay between correlated disorder and topology.
Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.