In recent experiments, time-dependent periodic fields are used to create exotic topological phases of matter with potential applications ranging from quantum transport to quantum computing. These non-equilibrium states, at high driving frequencies, exhibit the quintessential robustness against local disorder that is also seen at equilibrium. However, the existence of topological stability under a general setting is an open problem. We propose an effective theory that leverages on modern free probability theory and ideas in random matrices to analytically capture the existence of the topological phase for finite driving frequencies and across a range of disorder. We predict that, depending on the strength of disorder, such systems may be gapped (topological) or gapless (trivial). The theory can be used to predict properties of the transition between the two phases including the critical point. We corroborate our results by comparing to exact diagonalization for driven-disordered 1D Kitaev chain and 2D Bernevig-Hughes-Zhang (BHZ) models.
In recent experiments, time-dependent periodic fields are used to create exotic topological phases of matter with potential applications ranging from quantum transport to quantum computing. These non-equilibrium states, at high driving frequencies, exhibit the quintessential robustness against local disorder that is also seen at equilibrium. However, the existence of topological stability under a general setting is an open problem. We propose an effective theory that leverages on modern free probability theory and ideas in random matrices to analytically capture the existence of the topological phase for finite driving frequencies and across a range of disorder. We predict that, depending on the strength of disorder, such systems may be gapped (topological) or gapless (trivial). The theory can be used to predict properties of the transition between the two phases including the critical point. We corroborate our results by comparing to exact diagonalization for driven-disordered 1D Kitaev chain and 2D Bernevig-Hughes-Zhang (BHZ) models.
In recent experiments, time-dependent periodic fields are used to create exotic topological phases of matter with potential applications ranging from quantum transport to quantum computing. These nonequilibrium states, at high driving frequencies, exhibit the quintessential robustness against local disorder similar to equilibrium topological phases. However, proving the existence of such topological phases in a general setting is an open problem. We propose a universal effective theory that leverages on modern free probability theory and ideas in random matrices to analytically predict the existence of the topological phase for finite driving frequencies and across a range of disorder. We find that, depending on the strength of disorder, such systems may be topological or trivial and that there is a transition between the two. In particular, the theory predicts the critical point for the transition between the two phases and provides the critical exponents. We corroborate our results by comparing them to exact diagonalizations for driven-disordered 1D Kitaev chain and 2D Bernevig-Hughes-Zhang models and find excellent agreement. This Letter may guide the experimental efforts for exploring topological phases.
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.
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.
Topological phases represent a pillar of modern condensed matter physics. While gapped topological systems have been studied extensively, gapless topological materials represent an exciting, largely unexplored area. Here, the authors show that symmetry can enrich random quantum critical points and phases. They uncover a class of gapless topological phases, protected by symmetry and robust to strong randomness. Some of these phases can be realized in nonequilibrium states stabilized by many-body localization. They also appear naturally in periodically driven (Floquet) systems.
Periodically driven systems, also known as Floquet systems, can realize symmetry protected topological (SPT) phases that cannot be found in equilibrium. Here, we seek to understand the effects of strong disorder on such SPT phases, working with a one-dimensional Floquet system belonging to Altland-Zirnbauer class BDI. Using the transfer matrix method, we find that the disordered system hosts an array of trivial, topological, and Floquet topological phases. We then explore the phase diagram in the case of uniformly distributed bonds. Our analytic results are confirmed by a numeric computation of the bulk topological invariants. Although all states are generically localized, near phase transitions the localization length of topological edge states diverges. We derive the critical exponents governing these divergences, finding that they take on equilibrium values, despite the fact that we are considering phases unique to non-equilibrium. We also discuss how our work sets the stage for studying interacting Floquet SPT phases.
Topologically protected edge states exactly at topological phase boundaries challenge the conventional belief that topological states must be associated with a bulk energy gap. Because periodically driven (Floquet) systems host unusually intricate topological phase boundaries, topological edge states can be prolific at such Floquet quantum criticality. Working on a class of chiral-symmetric, Floquet-driven Majorana fermion chains, we analytically and computationally show that the precise boundaries between different Floquet topological gapped phases can accommodate topological edge modes, including the so-called critical Majorana π modes. We also identify a general bulk-edge correspondence formula to predict and understand the emergence of topological edge modes at Floquet quantum criticality. Of direct interest to quantum simulation experiments, our work opens an avenue for exploring topological physics intrinsic to nonequilibrium phase boundaries. The authors develop a theoretical framework to characterize the topology and edge states at the nonequilibrium critical points of periodically driven (Floquet) quantum systems. The rule of bulk-edge correspondence applicable to both gapped and gapless Floquet topological phases in 1D chiral symmetric driven systems is proposed.