Spectral gap of the antiferromagnetic Lipkin–Meshkov–Glick model

We study the spectral property of the supersymmetric (SUSY) antiferromagnetic Lipkin–Meshkov–Glick (LMG) model with an even number of spins and explicitly construct the supercharges of the model. Using the exact form of the SUSY ground state, we introduce simple trial variational states for the first excited states. We show 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, but because it is an upper bound, it does not allow rigorously determining whether the model is gapped or gapless. To answer this question, we obtain a nontrivial lower bound for the spectral gap and thus show that the antiferromagnetic SUSY LMG model is gapped for any even number of spins.