Exact spectrum of the Lipkin-Meshkov-Glick model in the thermodynamic limit and finite-size corrections

TL;DR

This paper provides an exact derivation of the Lipkin-Meshkov-Glick model's spectrum in the thermodynamic limit, analyzes finite-size corrections, and explores semi-classical expectation values of spin operators.

Contribution

It introduces an exact spectral analysis using spin coherent states, classifies spectral critical points, and computes finite-size and semi-classical effects.

Findings

Exact spectrum in the thermodynamic limit derived

Finite-size corrections include logarithmic terms

Semi-classical expectation values reveal subtle effects

Abstract

The spectrum of the Lipkin-Meshkov-Glick model is exactly derived in the thermodynamic limit by means of a spin coherent states formalism. In the first step, a classical analysis allows one to distinguish between four distinct regions in the parameter space according to the nature of the singularities arising in the classical energy surface; these correspond to spectral critical points. The eigenfunctions are then analyzed more precisely in terms of the associated roots of the Majorana polynomial, leading to exact expressions for the density of states in the thermodynamic limit. Finite-size effects are also analyzed, leading in particular to logarithmic corrections near the singularities occuring in the spectrum. Finally, we also compute expectation values of the spin operators in a semi-classical analysis in order to illustrate some subtle effects occuring in one region of the parameter space.