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

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 Schrödinger 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 [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] 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 behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.