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Physical Review A

Decay of quantum Loschmidt echo and fidelity in the broken phase of the Lipkin-Meshkov-Glick model

Qian Wang, Ping Wang, Yinbiao Yang, Wen-ge Wang

Published Apr 6, 2015• 10 citations
Publisher

Abstract

Analytical expressions are derived for the fidelity and for the quantum Loschmidt echo in the broken phase of the Lipkin-Meshkov-Glick (LMG) model for large particle number $N$. A detailed comparison is given between this model and some other models for the decaying behaviors of the two quantities in the neighborhood of critical points. The two quantities show qualitatively different decaying behaviors on the two sides of the critical point. Specifically, in the broken phase they behave in a way similar to those in models such as the one-dimensional Ising chain in a transverse field. In contrast, in the symmetric phase, the similarity is to those in models such as the Dicke model.

DOI
10.1103/physreva.91.042102

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Analytical expressions are derived for the fidelity and for the quantum Loschmidt echo in the broken phase of the Lipkin-Meshkov-Glick (LMG) model for large particle number NNN. A detailed comparison is given between this model and some other models for the decaying behaviors of the two quantities in the neighborhood of critical points. The two quantities show qualitatively different decaying behaviors on the two sides of the critical point. Specifically, in the broken phase they behave in a way similar to those in models such as the one-dimensional Ising chain in a transverse field. In contrast, in the symmetric phase, the similarity is to those in models such as the Dicke model.
In this paper, we study the scaling behavior of the ground-state logarithmic fidelity in the Lipkin–Meshkov–Glick (LMG) model. We find that the logarithmic fidelity shows a different scaling behavior in different phases of the model. It is an extensive quantity in the model's symmetry-broken phase while it behaves intensively in the polarized phase. Moreover, we also find that the logarithmic fidelity shows a singular behavior around the vicinity of the critical point and the critical exponent of the correlation length is also obtained numerically.
In this paper, we explore the differences between classical logarithmic fidelity and quantum fidelity. The classical logarithmic fidelity is found to be always extensive while the quantum one manifests distinct size dependence in different phases. Illustrated by the anisotropic Lipkin-Meshkov-Glick model, we found numerically and analytically that the logarithmic fidelity scales like N in the symmetry-broken phase and scales like N^0 in the polarized phase. The singular behavior around the critical point is also investigated.
We show that for quantum phase transitions with a single bosonic zero mode at the critical point, like the Dicke model and the Lipkin-Meshkov-Glick model, metric quantities such as fidelity, that is, the overlap between two ground states corresponding to two values ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ of the controlling parameter $\ensuremath{\lambda}$, depend only on the ratio $\ensuremath{\eta}=({\ensuremath{\lambda}}_{1}\ensuremath{-}{\ensuremath{\lambda}}_{c})/({\ensuremath{\lambda}}_{2}\ensuremath{-}{\ensuremath{\lambda}}_{c})$, where $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ at the critical point. This scaling property is valid also for time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.
We study the critical properties of the Lipkin-Meshkov-Glick model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytic case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are nontrivial results, indicating that the fidelity susceptibility is not always extensive.
The fidelity, linear entropy and entanglement entropy, concepts of information theory, are used to determine the localization of quantum phase transitions of the Lipkin–Meshkov–Glick (LMG) and the Dicke models for a finite number of particles. Another concept, the so-called fidelity susceptibility, gives us information of the scaling behavior of the quantum phase transitions of the models. In particular universal curves are presented for the specific susceptibility. It is shown that the control parameters of both models have a similar dependence in the number of particles when the systems approach to the thermodynamic limit.