Here, J > 0, with the sum indicated with i, j
means summing over i and j being neighboring sites
of the cubic lattice, and B is the externally applied
magnetic field, which we will assume is in the ˆz direction
for simplicity. The factor of 1/2 out front is
included so that each pair of spins is counted only
once. Each site i is assumed to have a spin Si of
spin S = 1/2. Here \mu B is the conventional Bohr