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Journal of Physics A Mathematical and General

Tunnelling at finite temperature in the LMG model

Alex H. Blin, Brigitte Hiller, Junqing Li

Published Jul 21, 1996• 2 citations
Publisher

Abstract

The Lipkin - Meshkov - Glick model is studied at finite temperature in mean field approximation. The imaginary time method is used to calculate the energy splitting due to tunnelling, by solving the equations of motion in Euclidean time and adapting the instanton approximation to finite temperatures. In this way we are also able to determine the pre-exponential factor at finite temperature.

DOI
10.1088/0305-4470/29/14/023

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The Lipkin - Meshkov - Glick model is studied at finite temperature in mean field approximation. The imaginary time method is used to calculate the energy splitting due to tunnelling, by solving the equations of motion in Euclidean time and adapting the instanton approximation to finite temperatures. In this way we are also able to determine the pre-exponential factor at finite temperature.
A consistent theory of the ground-state energy and its splitting due to the process of tunnelling for the Lipkin - Meshkov - Glick (LMG) model is presented. We calculate accurately the trivial and the instanton saddle-point contributions to the functional integral for the partition function of the model, in terms of the spin coherent states. We show that such a calculation has to be performed very accurately taking into account the discrete nature of the functional integral. This accurate consideration leads to the replacement of the magnitude of the spin s by s + 1/2, in the formula for the ground-state splitting obtained by a naive continuous method. We compare the numerical calculation of the ground-state energy and the splitting due to tunnelling with the results obtained by the quasiclassical method and obtain excellent agreement.
We investigate the thermal entanglement in the Lipkin—Meshkov—Glick (LMG) model which consists of spin-1/2 particles with XXZ-type exchange interactions between any pair of them. The ferromagnetic (FM) and antiferromagnetic (AFM) cases are completely analyzed at both finite temperature and zero temperature. According to the results obtained by accurate numerical calculation, several interesting physic phenomena and characteristics of thermal entanglement in the LMG model are found. Not only do we evaluate the entanglement of the LMG model, but also discover the correlations between macroscopic physical quantities and thermal entanglement.
The Lipkin-Meshkov-Glick (LMG) model originally describes a Fermionic many-body system in nuclear physics. However, in recent years, it has been widely found in condensed matter physics, quantum information systems, and quantum optics, and it is of wider and wider interest. Previous studies on this model mainly focused on the physics under the thermal dynamical limit, such as quantum phase transitions and quantum entanglement. There are also some researches about LMG model with finite size in some special limits, but the finite-size effect on energy spectrum is not very clear yet. This is the main motivation of this work. In this paper, the exact diagonalization method and the quantum perturbation theory are used to calculate and analyze the energy-level structure of the LMG model at a finite N. To solve it, we first map this model into the angular-momentum space to obtain a reduced LMG model. By this mapping, the dimension of Hilbert space is reduced to N+1 from 2N. The exact solution of its energy levels can be obtained easily in the U(1) limit where the total spin is conserved. We find that the levels are woven into a fishing-net structure in the U(1) limit. While away from the U(1) limit, the crossings between even and odd levels will open a gap, and the system's energy levels will be grouped into pairs with an odd and an even level, forming some bound states, called doublet states, and the parity of each doublet state will oscillate as the Zeeman field increases. This work gives the values of the critical Zeeman field for the parity crossings. These critical values shift as the interacting parameters and disappear at zero in the Z2 limit. In the Z2 limit, the system energy levels form splittings near the zero Zeeman field. In this article, we analytically calculate the relationship between these energy gaps and the Zeeman field. For odd and even number N, the parity of each state has a different behavior. Specifically, the ground state and the doublet excited states of the system with odd N will suffer a parity reversion at zero Zeeman field, while the states with even N will not. By tuning the interacting parameters, we also study the crossover from the U(1) limit to the Z2 limit. The parity oscillation we find in this system is a very important physical phenomenon, which also exists in some other systems like optical cavity quantum electrodynamics and magnetic molecule system.
The Lipkin-Meshkov-Glick model is used to examine the validity of some approximate methods in a many-body theory at finite temperatures. Namely, the thermal random phase approximation (TRPA) and the thermal renormalized random phase approximation (TRRPA) are studied. An average energy of the system, an average quasispin projection and a particle number variance are calculated within these approximation and exactly with the grand canonical ensemble partition function. On the whole the results of TRRPA are found to be in better agreement with the exact ones. The validity of the both approximation becomes better with increasing temperature as well as particle number.
We study the adiabatic dynamics of the Lipkin-Meshkov-Glick (LMG) model close to its quantum critical point by linearly switching the transverse field from an initial large value to zero. We concentrate our attention on the residual energy after the quench in order to characterize the level of diabaticity of the evolution. As a function of the characteristic time of the quench $\ensuremath{\tau}$ we identify three different regimes. For fast quenches the residual energy Eres{E}_{\text{res}}Eres​ is almost independent on $\ensuremath{\tau}$. For slower quenches a second intermediate region appears in which a powerlike decay emerges with ${E}_{\text{res}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}3/2}$. By further slowing the quench rate, we find a third large-$\ensuremath{\tau}$ regime characterized by a different power law, ${E}_{\text{res}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2}$. All these findings can be accounted for by means of an effective Landau-Zener approximation for the finite-size LMG model. We complete our description of the adiabatic dynamics of the LMG model through the analysis of the entanglement entropy of the evolved state.