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International Journal of Modern Physics B

Quench dynamics of edge modes revealed by entanglement spectrum in a dimerized chain with dilute impurities

Qiu-Lan Zhang

Published Jul 19, 2018• 0 citations
Publisher

Abstract

We study the evolution of the one-particle entanglement spectrum for the one-dimensional dimerized chain in the presence of impurities. When the system quenches from a topologically nontrivial phase to a trivial phase with disorder, the spectrum features the split of two zero-energy edge modes due to their interaction with impurities. Our observations demonstrate the feasibility of using entanglement spectrum to study the quench dynamics of topological systems.

DOI
10.1142/s0217979218502405

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We study the evolution of the one-particle entanglement spectrum for the one-dimensional dimerized chain in the presence of impurities. When the system quenches from a topologically nontrivial phase to a trivial phase with disorder, the spectrum features the split of two zero-energy edge modes due to their interaction with impurities. Our observations demonstrate the feasibility of using entanglement spectrum to study the quench dynamics of topological systems.
One of the appealing features of topological systems is the presence of robust edge modes. Under a sudden quantum quench, the edge modes survive for a characteristic time that scales with the system size, during which the nontrivial topology continues to manifest in entanglement properties, even though the post-quench Hamiltonian belongs to a trivial phase. We exemplify this in the quench dynamics of a two-dimensional Haldane model with the help of one-particle entanglement spectrum and the probability density of its mid-states. We find that, beyond our knowledge in one-dimensional models, the momentum dependence of the transverse velocity plays a crucial role in the out-of-equilibrium evolution of the entanglement properties.
We investigate the quench dynamics of the one-particle entanglement spectra (OPES) for systems with topologically nontrivial phases. By using dimerized chains as an example, it is demonstrated that the evolution of OPES for the quenched bipartite systems is governed by an effective Hamiltonian which is characterized by a pseudospin in a time-dependent pseudomagnetic field S(k,t). The existence and evolution of the topological maximally entangled states (tMESs) are determined by the winding number of S(k,t) in the k-space. In particular, the tMESs survive only if nontrivial Berry phases are induced by the winding of S(k,t). In the infinite-time limit the equilibrium OPES can be determined by an effective time-independent pseudomagnetic field Seff(k). Furthermore, when tMESs are unstable, they are destroyed by quasiparticles within a characteristic timescale in proportion to the system size.
Based on the results of exact analytic calculations, we show that topological edge states and impurities in quantum dimerized chains manifest themselves in various local static and dynamical characteristics, which can be measured in experiments. In particular, topological edge states can be observed in the magnetic field behavior of the local magnetization or magnetic susceptibility of dimerized spin chains as jumps (for the magnetization) and features (for the static susceptibility) at zero field. In contrast, impurities reveal themselves in similar jumps and features, however, at nonzero values of the critical field. We also show that dynamical characteristics of dimerized quantum chains also manifest the features, related to the topological edge states and impurities. Those features, as a rule, can be seen more sharply than the manifestation of bulk extended states in, e.g., the dynamical local susceptibility. Such peculiarities can be observed in one-dimensional dimerized spin chains, e.g., in NMR experiments, or in various realizations of quantum dimerized chains in optical experiments.
Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at (d−n)(d-n)(d−n)-dimensional edges, where n>1n > 1n>1. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.
We unveil the stable (d+1)-dimensional topological structures underlying the quench dynamics for all of the Altland-Zirnbauer classes in d=1 dimension, and we propose to detect such dynamical topology from the time evolution of entanglement spectra. Focusing on systems in classes BDI and D, we find crossings in single-particle entanglement spectra for quantum quenches between different symmetry-protected topological phases. The entanglement-spectrum crossings are shown to be stable against symmetry-preserving disorder and faithfully reflect both Z (class BDI) and Z_{2} (class D) topological characterizations. As a by-product, we unravel the topological origin of the global degeneracies temporarily emerging in the many-body entanglement spectrum in the quench dynamics of the transverse-field Ising model. These findings can experimentally be tested in ultracold atoms and trapped ions with the help of cutting-edge tomography for quantum many-body states. Our work paves the way towards a systematic understanding of the role of topology in quench dynamics.
We study the quench dynamics of entanglement spectra in the Kitaev chain with variable-range pairing quantified by power-law decay rate $\ensuremath{\alpha}$. Considering the post-quench Hamiltonians with flat bands, we demonstrate that the presence of entanglement-spectrum crossings during its dynamics is able to characterize the topological phase transitions (TPTs) in both short-range ($\ensuremath{\alpha}>1$) and long-range ($\ensuremath{\alpha}<1$) sectors. Properties of entanglement-spectrum dynamics are revealed for the quench protocols in the long-range sector or with $\ensuremath{\alpha}$ as the quench parameter. In particular, when the lowest upper-half entanglement-spectrum value of the initial Hamiltonian is smaller than the final one, the TPTs can also be diagnosed by the difference between the lowest two upper-half entanglement-spectrum values if the halfway winding number is not equal to that of the initial Hamiltonian. Moreover, we discuss the stability of characterizing the TPTs via entanglement-spectrum crossings against energy dispersion in the long-range model.