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arXiv:1705.04547 · Physical review. E

Probing the excited-state quantum phase transition through statistics of Loschmidt echo and quantum work

Qian Wang, H. T. Quan

Published Sep 28, 2017• 35 citations
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Abstract

By analyzing the probability distributions of the Loschmidt echo (LE) and quantum work, we examine the nonequilibrium effects of a quantum many-body system, which exhibits an excited-state quantum phase transition (ESQPT). We find that depending on the value of the controlling parameter the distribution of the LE displays different patterns. At the critical point of the ESQPT, both the averaged LE and the averaged work show a cusplike shape. Furthermore, by employing the finite-size scaling analysis of the averaged work, we obtain the critical exponent of the ESQPT. Finally, we show that at the critical point of ESQPT the eigenstate is a highly localized state, further highlighting the influence of the ESQPT on the properties of the many-body system.

DOI
10.1103/physreve.96.032142

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By analyzing the probability distributions of the Loschmidt echo (LE) and quantum work, we examine the nonequilibrium effects of a quantum many-body system, which exhibits an excited-state quantum phase transition (ESQPT). We find that depending on the value of the controlling parameter the distribution of the LE displays different patterns. At the critical point of the ESQPT, both the averaged LE and the averaged work show a cusplike shape. Furthermore, by employing the finite-size scaling analysis of the averaged work, we obtain the critical exponent of the ESQPT. Finally, we show that at the critical point of ESQPT the eigenstate is a highly localized state, further highlighting the influence of the ESQPT on the properties of the many-body system.
Identifying dynamical signatures of excited-state quantum phase transitions (ESQPTs) in experimentally realizable quantum many-body systems is helpful for understanding the dynamical effects of ESQPTs. In such systems, the highly controllable spinor Bose-Einstein condensates (BECs) offer an exceptional platform to study ESQPTs. In this work, we investigate the dynamical characteristics of the ESQPT in spin-1 BEC by means of the Loschmidt echo spectrum. The Loschmidt echo spectrum is an extension of the well-known Loschmidt echo and defined as the overlaps between the evolved state and the excited states of the initial Hamiltonian. We show that both the time-evolved and long-time-averaged Loschmidt echo spectrum undergo a remarkable change as the system passes through the critical point of the ESQPT. Moreover, the particular behavior exhibited by the Loschmidt echo spectrum at the critical point stands as a dynamical detector for probing the ESQPT. We further demonstrate how to capture the features of the ESQPT by using the energy distribution associated with the Loschmidt echo spectrum for time-evolved and long-time-averaged cases, respectively. Our findings contribute to a further verification of the usefulness of the Loschmidt echo spectrum for witnessing various quantum phase transitions in many-body systems and provide a different way to experimentally examine the dynamical consequences of ESQPTs.
Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited-state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by the dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.
Studying the implications and characterizations of the excited-state quantum phase transitions (ESQPTs) would enable us to understand various phenomena observed in quantum many-body systems. In this work, we delve into the affects and characterizations of the ESQPTs in the anharmonic Lipkin-Meshkov-Glick (LMG) model by means of the entropy of the quantum work distribution. The entropy of the work distribution measures the complexity of the work distribution and behaves as a valuable tool for analyzing nonequilibrium work statistics. We show that the entropy of the work distribution captures salient signatures of the underlying ESQPTs in the model. In particular, a detailed analysis of the scaling behavior of the entropy verifies that it not only acts as a witness of the ESQPTs but also reveals the difference between different types of ESQPTs. We further demonstrate that the work distribution entropy also behaves as a powerful tool for understanding the features and differences of ESQPTs in the energy space. Our results provide further evidence of the usefulness of the entropy of the work distribution for investigating various phase transitions in quantum many-body systems and open up a promising way for experimentally exploring the signatures of ESQPTs.
Excited-state quantum phase transitions (ESQPTs) strongly influence the spectral properties of collective many-body quantum systems, changing degeneracy patterns in different quantum phases. Level degeneracies in turn affect the system's dynamics. We analyze the degeneracy dependence on the size of two-level boson models with a u(n+1)u(n+1)u(n+1) dynamical algebra, where nnn is the number of collective degrees of freedom. Below the ESQPT critical energy of these models, the energy gap between neighboring levels that belong to different symmetry sectors gets close to zero as the system size increases. We report and explain why this gap goes to zero exponentially for systems with one collective degree of freedom but algebraically in models with more than one degree of freedom. As a consequence, we show that the infinite-time average of out-of-time-order correlators is an ESQPT order parameter in finite systems with n=1n=1n=1, but in systems with n>1n>1n>1, this average only works as an order parameter in the mean-field limit.
Abstract We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanced decay behavior of Loschmidt echo. Our studies show how to detect equilibrium many-body phase transitions with biorthogonal Loschmidt echo that can be observed in future experiments via quantum dynamics after a quench.
Multifractality of quantum states plays an important role for understanding numerous complex phenomena observed in different branches of physics. The multifractal properties of the eigenstates allow for characterising various phase transitions. In this work, we perform a thorough analysis of the impacts of an excited-state quantum phase transition (ESQPT) on the fractal behavior of both static and dynamical wave functions in a ferromagentic spin-1 Bose-Einstein condensate. By studying the features of the fractal dimensions, we show how the multifractality of eigenstates and time evolved states are affected by the presence of ESQPT. Specifically, the underlying ESQPT leads to a strong localization effect, which in turn enables us to use it as an indicator of ESQPT. We verify the ability of the fractal dimensions to probe the occurrence of ESQPT through a detailed scaling analysis. We also discuss how the ESQPT manifests itself in the fractal dimensions of the long-time averaged state. Our findings further confirm that the multifractal analysis is a powerful tool for studying of phase transitions in quantum many-body systems and also hint an potential application of ESQPTs in the burgeoning field of state preparation engineering.