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

• Main Authors: Co', Giampaolo, De Leo, Stefano
• Format: Journal Article
• Language: English
• Published: 31-05-2018
• Subjects: Physics - Nuclear Theory; Physics - Quantum Physics
• DOI: 10.48550/arxiv.1805.12442

International Journal of Modern Physics E 27, 1850039-18 (2018) 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.