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arXiv:cond-mat/0102073 · Journal of Physics A Mathematical and General

Instanton picture of the spin tunnelling in the Lipkin - Meshkov - Glick model

V I Belinicher, C Providencia, João da Providência

Published Aug 21, 1997• 21 citations

Abstract

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.

DOI: 10.1088/0305-4470/30/16/008

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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 analyze the finite-size scaling exponents in the Lipkin-Meshkov-Glick model by means of the Holstein-Primakoff representation of the spin operators and the continuous unitary transformations method. This combination allows us to exactly compute the leading corrections to the ground-state energy, the gap, the magnetization, and the two-spin correlation functions. We also present numerical calculations for large system size which confirm the validity of this approach. Finally, we use these results to discuss the entanglement properties of the ground state focusing on the (rescaled) concurrence that we compute in the thermodynamical limit.

We study the spectral property of the supersymmetric (SUSY) antiferromagnetic Lipkin–Meshkov–Glick (LMG) model with an even number of spins and explicitly construct the supercharges of the model. Using the exact form of the SUSY ground state, we introduce simple trial variational states for the first excited states. We show numerically that they provide a relatively accurate upper bound for the spectral gap (the energy difference between the ground state and first excited states) in all parameter ranges, but because it is an upper bound, it does not allow rigorously determining whether the model is gapped or gapless. To answer this question, we obtain a nontrivial lower bound for the spectral gap and thus show that the antiferromagnetic SUSY LMG model is gapped for any even number of spins.

Originally introduced in nuclear physics as a numerical laboratory to test different many-body approximation methods, the Lipkin-Meshkov-Glick (LMG) model has received much attention as a simple enough but non-trivial model with many interesting features for areas of physics beyond the nuclear one. In this contribution we look at the LMG model as a particular example of an SU (1,1) Richardson-Gaudin model. The characteristics of the model are analyzed in terms of the behavior of the spectral-parameters or pairons which determine both eigenvalues and eigenfunctions of the model Hamiltonian. The problem of finding these pairons is mathematically equivalent to obtain the equilibrium positions of a set of electric charges moving in a two dimensional space. The electrostatic problems for the different regions of the model parameter space are discussed and linked to the different energy density of states already identified in the LMG spectrum.