Analytical and numerical analysis of the complete Lipkin-Meshkov-Glick Hamiltonian

TL;DR

This paper provides an analytical and numerical study of the Lipkin-Meshkov-Glick model, highlighting how the inclusion of the W interaction term affects the energy spectrum, degeneracies, and energy gaps in quantum many-body systems.

Contribution

It offers new analytical solutions for small systems and explores the impact of the W interaction on the model's spectral properties, extending understanding beyond previous approximations.

Findings

W interaction alters eigenstate degeneracies

Energy gap between ground and first excited state is affected

Analytical solutions for systems up to eight particles are provided

Abstract

The Lipkin-Meshkov-Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schr\"odinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the $V$ interaction, which promotes or degrade pairs of particles, and a second one, the $W$ interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the $W$ term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behaviour can be well understood by considering the extrapolations of the analytical results. Of particular interest it is the study of how the $W$ interaction affects the energy gap between the ground state and the first-excited state.