Spectral Gap of the Anti-Ferromagnetic Lipkin-Meshkov-Glick Model

Abstract

The spectral property of the supersymmetric (SUSY) antiferromagnetic Lipkin-Meshkov-Glick (LMG) model with an even number of spins is studied. The supercharges of the model are explicitly constructed. By using the exact form of the supersymmetric ground state we introduce simple trial variational states for first excited states. It is demonstrated 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. However, being an upper bound, it does not allow us to determine vigorously whether the model is gapped or gapless. Here, we provide a non-trivial lower bound for the spectral gap and thereby show that the antiferromagnetic SUSY LMG model is gapped for any even number of spins.