We calculate the order parameter m in the infinite-range isotropic-interaction quantum model of antiferromagnetism due to Lieb and Mattis (LM). The Hamiltonian H1 in this model contains pure Heisenberg interactions. m is the mean value of the z component msz of the staggered spin per particle ms in the ground state of H1 plus a symmetry-breaking term HB due to an infinitesimal staggered field B in the thermodynamic limit (TL). We find a value of √3 for the ratio r=m/m0, m0 being the root-mean-square value of msz for B==0 in the TL. m and m0 measure, respectively, the spontaneous symmetry breaking and the amount of long-range order in the symmetry-unbroken state. This value is expected on the basis of our recent argument that r= √3 if ms behaves in the TL as a classical vector with no magnitude fluctuations—behavior that we show holds for the Lieb-Mattis model. The addition to H1 of a small easy-axis anisotropic term is also discussed briefly. These studies allow calculation of the size dependence of quantities that we have recently shown to become lower bounds on m in the TL (bounds that are valid for short-range-interaction models as well). In the LM model, low-lying energy eigenstates with excitation energies of order N-1, where N is the number of spins, form a continuum (with no gap) in the TL. These are closely related to similar states shown to exist for the nearest-neighbor isotropic Heisenberg antiferromagnet. We show that this continuum has a density of states that is O(N0); i.e., it is not extensive.
Kaplan, T. A., von der Linden, W., & Horsch, P. (1990). Spontaneous symmetry breaking in the Lieb-Mattis model of antiferromagnetism. Physical Review / B, 42(7), 4663. https://doi.org/10.1103/PhysRevB.42.4663