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arXiv:1610.00851 · Journal of Statistical Mechanics Theory and Experiment

Relaxation of the entanglement spectrum in quench dynamics of topological systems

Yi-Hao Jhu, Pochung Chen, Ming-Chiang Chung

Published Jul 1, 2017• 5 citations
arXivHTMLPDF

Abstract

Abstract In this paper, we investigate how the entanglement spectrum relaxes to its steady-state values in one-dimensional quadratic systems after a quantum quench. In particular, we apply saddle-point expansion to dimerized chains and 1D p -wave superconductors. We found the entanglement spectrum to always exhibit a power-law relaxation superimposed with oscillations at certain characteristic angular frequencies. For dimerized chains, we found the exponent ν of the power-law decay to always be 3 / 2 . For 1D p -wave superconductors, however, we found that, depending on the initial and final Hamiltonian, the exponent ν can take its value from a limited list of values, the smallest possible of which is ν = 1 / 2 , which leads to a very slow convergence to its steady-state value.

DOI
10.1088/1742-5468/aa79b3

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Abstract In this paper, we investigate how the entanglement spectrum relaxes to its steady-state values in one-dimensional quadratic systems after a quantum quench. In particular, we apply saddle-point expansion to dimerized chains and 1D p -wave superconductors. We found the entanglement spectrum to always exhibit a power-law relaxation superimposed with oscillations at certain characteristic angular frequencies. For dimerized chains, we found the exponent ν of the power-law decay to always be 3 / 2 . For 1D p -wave superconductors, however, we found that, depending on the initial and final Hamiltonian, the exponent ν can take its value from a limited list of values, the smallest possible of which is ν = 1 / 2 , which leads to a very slow convergence to its steady-state value.
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.
We study the universal dynamical relaxation behaviors of a quantum XY chain following a quench, paying special attention to the case in which the initial state is a critical ground state, or the postquenched Hamiltonian is at a critical point of equilibrium quantum phase transition, or both of them are critical. In such a ``critical quench,'' we find very interesting real-time dynamical scaling behaviors and we find crossover phenomena between them. For a quench from a noncritical point to a critical point, we find that, compared to the noncritical quench, the universal power-law scaling behavior does not change; however, there may be a crossover between the exponential decaying behavior and the power-law scaling. For a quench from a critical point to a noncritical point, the power-law scaling behaviors ${t}^{\ensuremath{-}3/2}$ and ${t}^{\ensuremath{-}3/4}$ in the noncritical quenches may be changed to ${t}^{\ensuremath{-}1}$ and ${t}^{\ensuremath{-}1/2}$, respectively. If the prequenched Hamiltonian is set to be a point that is close to but not exactly at a critical point, we find interesting crossover phenomena between different power-law scaling behaviors. We also study the quench from the vicinity of a multicritical point, we find crossover behaviors that are induced by a different mechanism, and we find another crossover exponent. All the results are related to the gap-closing properties of the energy spectrum of the critical points.
We study the universal real-time relaxation behaviors of a long-range quantum XY chain following a quench. Our research includes both the noncritical and critical quench. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of t−3/2t^{-3/2}t−3/2 or t−1/2t^{-1/2}t−1/2, respectively, which is the same as the counterpart of the short-range XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law t−μt^{-μ}t−μ may be different from the t−3/4t^{-3/4}t−3/4 law of the counterpart of the short-range model. More interestingly, the decaying exponent μμμ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law t−μt^{-μ}t−μ may be changed because of the gap-closing property of the energy spectrum of the critical point.
We investigate the time evolution towards the asymptotic steady state of a one-dimensional interacting system after a quantum quench. We show that at finite times the latter induces entanglement between right- and left-moving density excitations, encoded in their cross-correlators, which vanishes in the long-time limit. This behavior results in a universal time decay $\ensuremath{\propto}{t}^{\ensuremath{-}2}$ of the system spectral properties, in addition to nonuniversal power-law contributions typical of Luttinger liquids. Importantly, we argue that the presence of quench-induced entanglement clearly emerges in transport properties, such as charge and energy currents injected in the system from a biased probe and determines their long-time dynamics. In particular, the energy fractionalization phenomenon turns out to be a promising platform to observe the universal power-law decay $\ensuremath{\propto}{t}^{\ensuremath{-}2}$ induced by entanglement and represents a novel way to study the corresponding relaxation mechanism.
Abstract We investigate the relaxation dynamics of the fermion two-point correlation function C mn ( t ) = 〈 ψ ( t ) ∣ c m † c n ∣ ψ ( t ) 〉 in the XY chain with alternating nearest-neighbor hopping interaction after a quench. We find that the deviation δ C mn (t) = C mn (t) − C mn (∞) decays with time following the power law behavior t −μ , where the exponent μ depends on whether the quench is to the commensurate phase (μ = 1) or incommensurate phase ( μ = 1 2 ). This decay of δ C mn (t) arises from the transient behavior of the double-excited quasiparticle occupations and the transitions between different excitation spectra. Furthermore, we find that the steady value C mn (∞) only involves the average fermion occupation numbers (i.e. the average excited single particle) over the time-evolved state, which are different from the ground state expectation values. We also observe nonanalytic singularities in the steady value C mn (∞) for the quench to the critical points of the quantum phase transitions (QPTs), suggesting its potential use as a signature of QPTs.