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Physical review. B./Physical review. B

Dynamical topological quantum phase transitions in high-order topological systems

Hai-Xiao Xiao, Weilun Jiang, Peng Qian, Hongju Li, Zhongjun Li, Heng Shen, Bing Chen

Published Aug 7, 2024• 9 citations

Abstract

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.

DOI: 10.1103/physrevb.110.064306

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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.

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.

Topological characteristics of quantum systems are typically determined by the closing of a gap, while the dynamical quantum phase transition (DQPT) during quantum real-time evolution has emerged as a nonequilibrium analog to the quantum phase transition (QPT). In this paper, we illustrate that the system dynamics can be elucidated by considering the precession of a collection of free pseudospins under a magnetic field based on the exact results of extended Kitaev chains. The topology of the driven Hamiltonian is determined by the average winding number of the nonequilibrium state. Furthermore, we establish that the singularity of the DQPT arises from two perpendicular pseudospin vectors associated with the pre- and postquenched Hamiltonians. Moreover, we investigate the distinct behaviors of the dynamic pairing order parameter in both topological and nontopological regions. These findings offer valuable insights into the non-equilibrium behavior of topological superconductors, contributing to the understanding of the resilience of topological properties in driven quantum systems.

Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.

We focus on the dynamical characters for a topological superconductor model. Considering a chiral p-wave superconductor with open and periodic boundary conditions in orthogonal directions, we show the bulk-boundary correspondence comparing to the topological phase diagram by Pfaffian. The expectations of Pauli matrixes are used to reflect the topological phase for this model. The dynamical quantum phase transition revealed by quench does occur in the parameter region consistent with the topological phase diagram. Besides quenching, temporal dynamical non-analyticities occur until the parameter crossing the topological critical point slowly. And the dynamics of the excitation behave differently in different topological phases. The dynamical phase transition behaves robustly against noise during evolution until the amplitude of noise reaching a threshold with the time-Anderson localization dominates. This work manifests the topological phase can be detected in a dynamical way.

Dynamical quantum phase transitions (DQPT) provide a fundamental framework for characterizing nonequilibrium critical phenomena, and have recently been extended to non-Hermitian systems. Half-integer jumps of the dynamical topological order parameter can arise in both Hermitian and non-Hermitian systems, yet their origin in non-Hermitian settings remains unclear. Here, we investigate DQPTs in a two-band non-Hermitian model and show that these fractional jumps originate from the discontinuity in the complex energy spectrum. We demonstrate that this behavior is intrinsically linked to the steady-state phase diagram of the quench Hamiltonian, and also classify the occurrence of DQPT. We further extend our analysis to the multiband Hamiltonian and the two-dimensional model, providing a unified understanding of fractional dynamical topological signatures.

We study the nonequilibrium time evolution of a variety of one-dimensional (1D) and two-dimensional (2D) systems (including SSH model, Kitaev-chain, Haldane model,

p+ip

superconductor, etc.) following a sudden quench. We prove analytically that topology-changing quenches are always followed by nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo), referred to as dynamical phase transitions (DPTs) in the literature. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of topologically protected nonequilibrium time scales are determined by the difference between the initial and final winding numbers, while in 2D systems no such relation exists for the Chern numbers. The singularities of dynamical free energy in the 2D case are qualitatively different from those of the 1D case; the cusps appear only in the first time derivative.