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.
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.
Abstract Topological phases of matter can appear in noninteracting systems, as in band topology, or interacting systems, such as in spin models, with their defining features typically robust against weak disorder. Intriguingly, disorder itself can also induce topological phases--exemplified by the Anderson topological insulator in noninteracting systems. Experimental studies on disorder-induced topology have so far been limited to band topology. Here we report direct observations of disorder-induced many-body interacting average topological order in an atom array at half-filling, whereby random offsets to tweezer locations forming a lattice implement structural disorder, causing fluctuating long-range dipolar interactions between tweezer confined single atoms. The ground state degeneracy in disordered configurations is detected and compared to the regular lattice. The induced topological phase is vindicated by the spatially resolved atom-atom correlation functions for different forms of dimer compositions. By probing the quench dynamics of a highly excited state, we observe markedly slower decay of edge spin magnetization in comparison to the bulk spin, consistent with the presence of topologically protected edge modes in disordered lattices. Our experiments open a new direction for studying the interplay between disorder and strongly interacting topological matter in Rydberg atom arrays.
Topological states of matter in disordered systems without translation symmetry have attracted great interest in recent years. These states with topological characters are not only robust against certain disorders, but also can be counterintuitively induced by disorders from a topologically trivial phase in the clean limit. In this review, we summarize the current theoretical and experimental progress on disorder-induced topological phases in both condensed-matter and artificial systems. We first introduce the topological Anderson insulators (TAIs) induced by random disorders and their topological characterizations and experimental realizations. We then discuss various extensions of TAIs with unique localization phenomena in quasiperiodic and non-Hermitian systems. We also review the theoretical and experimental studies on the disorder-induced topology in dynamical and many-body systems, including topological Anderson-Thouless pumps, disordered correlated topological insulators and average-symmetry protected topological orders acting as interacting TAI phases. Finally, we conclude the review by highlighting potential directions for future explorations.
The topological Anderson and Mott insulators are two phases that have so far been separately and widely explored beyond topological band insulators. Here we combine the two seemingly different topological phases into a system of spin-1/2 interacting fermionic atoms in a disordered optical lattice. We find that the topological Anderson and Mott insulators in the noninteracting and clean limits can be adiabatically connected without gap closing in the phase diagram of our model. Lying between the two phases, we uncover a disordered correlated topological insulator, which is induced from a trivial band insulator by the combination of disorder and interaction, as the generalization of topological Anderson insulators to the many-body interacting regime. The phase diagram is determined by computing various topological properties and confirmed by unsupervised and automated machine learning. We develop an approach to provide a unified and clear description of topological phase transitions driven by interaction and disorder. The topological phases can be detected from disorder-/interaction-induced edge excitations and charge pumping in optical lattices.
While the effects of off-diagonal disorder have been extensively studied in low-dimensional topological systems, such research remains insufficient in four-dimensional (4D) topological systems. In this work, we investigate the effect of off-diagonal disorders on a 4D topological insulator. We find that off-diagonal disorder can induce a 4D topological Anderson insulator (TAI) phase. This differs from previous finding in two-dimensional topological insulators, where off-diagonal disorder fails to induce the TAI phase. Specifically, we consider 16 different configurations of disorder and identify two distinct types based on their effect on the system. Furthermore, using the self-consistent Born approximation, we elucidate the unique role of off-diagonal disorder in these phase transitions compared to the diagonal Anderson disorder.
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.
The long-range Su-Schrieffer-Heeger (SSH) model, in which the second nearest-neighbor hopping is taken into account, exhibits a topological phase diagram that contains winding numbers w=0,1, and 2. In the clean system, the change in winding number stems from the band-touching phenomenon. In the presence of disorder, the renormalization of energy band and Fermi level results in the nonzero density of states in the energy gap. These midgap states cause the crossover phenomenon and the divergence of localization length at a critical disorder strength Uc in the finite SSH system. In this study, we numerically computed the mean winding number and localization length for the disordered SSH system. We find that the disorder is able to drive phase transitions between different mean winding numbers: $w=0\ensuremath{\rightarrow}1$, $0\ensuremath{\rightarrow}2$, $1\ensuremath{\rightarrow}2$, and $2\ensuremath{\rightarrow}1$ in the weak disorder regime. By investigating the wave function distribution and the self-energy, the nonzero mean winding numbers correspond to the so-called topological Anderson insulating (TAI) phases. The finite size scaling for the mean winding number in the TAI phase is shown. For describing the phase transitions in the thermodynamic limit, we apply the criterion of band gap closure resulting from the broadening of energy band and Fermi level to determine the critical disorder strength. The critical disorder strength for self-consistent Born approximation (SCBA) UcSCBA and that for first Born approximation (FBA) UcFBA are numerically calculated. UcSCBA is found to match with Uc qualitatively. Nonetheless, SCBA indicates the different roles of band shifts and Fermi level broadening near the topological phase transitions. Band shift/Fermi level broadening is more dominant for the transitions from low-to-high/high-to-low winding number. Interestingly, for the transition from bulk insulator to TAI, UcFBA is quantitatively closer to Uc than UcSCBA as long as the renormalized band gap is zero within FBA.