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arXiv:1909.01402 · Physical review. B./Physical review. B

Stability of dynamical quantum phase transitions in quenched topological insulators: From multiband to disordered systems

Christian B. Mendl, Jan Carl Budich

Published Dec 26, 2019• 26 citations
arXivHTMLPDF

Abstract

Dynamical quantum phase transitions (DQPTs) represent a counterpart in nonequilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multiorbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions and construct basic counterexamples to the occurrence of DQPTs in multiband topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multiband to disordered systems, we approach random disorder by increasing the (super)unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.

DOI
10.1103/physrevb.100.224307

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Dynamical quantum phase transitions (DQPTs) represent a counterpart in nonequilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multiorbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions and construct basic counterexamples to the occurrence of DQPTs in multiband topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multiband to disordered systems, we approach random disorder by increasing the (super)unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.
We investigate the nonequilibrium dynamics of two multiband boundary-obstructed topological systems (BOTSs) and a higher-order topological superconductor (HOTSC). The singular behavior exhibited by the dynamical free energy indicates the presence of dynamical quantum phase transitions (DQPTs) within these systems. By studying the nonequilibrium dynamics in the BOTS featuring even-parity Cooper pairing, we find that there are DQPTs in this BOTS via quenching from different topological phase regions. The analysis of the energy spectra shows that two previously considered identical topological phases are actually different. Based on this result, we identify that all DQPTs in the BOTS featuring even-parity Cooper pairing result from topology-changing quenches. Moreover, we also study DQPTs in another BOTS featuring odd-parity Cooper pairing, and we obtain similar results to those from the BOTS featuring even-parity Cooper pairing. The appearance of DQPTs in different BOTSs reveals that the occurrence of DQPTs is related to the topology of BOTSs. In addition, we adopt the same method to investigate DQPT in HOTSC for comparison. The numerical results show that DQPT always takes place for quenches between topological and trivial phases. This is completely different from the results of BOTSs, for which not all topology-changing quenches are accompanied by DQPTs. The study of DQPTs unveils the apparent difference between BOTS and HOTSC, and thus our study provides an alternative approach to revealing the topological nature of novel higher-order topological systems, which does not rely on the specific definitions of topological invariants.
Understanding the properties of far-from-equilibrium quantum systems is becoming a major challenge of both fundamental and applied physics. For instance, the lack of thermalization in integrable and (many body) localized systems provides new insights in the understanding of the relaxation dynamics of quantum phases. On a more applicative side, the possibility of exploiting the properties of far-from-equilibrium states, for example in pump-probe experiments, opens unprecedented scenarios. The effort in providing a classification of far-from-equilibrium phases, in terms of local or topological order parameters, is hence intense. In this context, the concept of Dynamical Quantum Phase Transition (DQPT) has been introduced. A DQPT is (roughly) defined as a zero of the Loschmidt-Echo as a function of time and represents a natural non-equilibrium counterpart of a thermal phase transition. Here, we investigate the DQPTs occurring in the quantum xy chain subject to a quantum quench of finite duration. We show that the number of distinct DQPTs can vary as the duration of the quantum quench is varied. However, the parity of such number only depends on the pre-quench and post-quench Hamiltonians and is related to a topological invariant.
Dynamical quantum phase transitions (DQPTs), which refer to the criticality in time of a quantum many-body system, have attracted much theoretical and experimental research interest recently. Despite that DQPTs are defined and signaled by the nonanalyticities in the Loschmidt rate, its interrelation with various correlation measures such as the equilibrium order parameters of the system remains unclear. In this work, by considering the quench dynamics in an interacting topological model, we find that the equilibrium order parameters of the model in general exhibit signatures around the DQPT, in the short time regime. The first extrema of the equilibrium order parameters are connected to the first Loschmidt rate peak. By studying the unequal-time two-point correlation, we also find that the correlation between the nearest neighbors decays while that with neighbors further away builds up as time grows in the noninteracting case, and upon the addition of repulsive intracell interactions. On the other hand, the intercell interaction tends to suppress the two-site correlations. These findings could provide us insights into the characteristic of the system around DQPTs, and pave the way to a better understanding of the dynamics in nonequilibrium quantum many-body systems.
We introduce and study the dynamical probes of band-structure topology in the postquench time evolution of quantum many-body systems initialized in mixed states. Our construction generalizes the notion of dynamical quantum phase transitions (DQPTs), a real-time counterpart of conventional equilibrium phase transitions in quantum dynamics, to finite temperatures and generalized Gibbs ensembles. The nonanalytical signatures hallmarking these mixed-state DQPTs are found to be characterized by observable phase singularities manifesting in the dynamical formation of vortex-antivortex pairs in the interferometric phase of the density matrix. Studying quenches in Chern insulators, we find that changes in the topological properties of the Hamiltonian can be identified in this scenario, without ever preparing a topologically nontrivial or low-temperature initial state. Our observations are of immediate relevance for current experiments aimed at realizing topological phases in ultracold atomic gases.
A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state-i.e., the Loschmidt echo-vanishes at critical times {t^{*}}. Analytical results to date are limited to two-band models, leaving the exact relation between topology and DQPT unclear. In this Letter, we show that, for a general multiband system, a robust DQPT relies on the existence of nodes (i.e., zeros) in the wave function overlap between the initial band and the postquench energy eigenstates. These nodes are topologically protected if the two participating wave functions have distinctive topological indices. We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model. We also discuss possible experimental observations.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.