  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.
