Spontaneous symmetry breaking in the Lieb-Mattis model of antiferromagnetism

We calculate the order parameter m in the infinite-range isotropic-interaction quantum model of antiferromagnetism due to Lieb and Mattis (LM). The Hamiltonian ${\mathit{H}}_1$ in this model contains pure Heisenberg interactions. m is the mean value of the z component ${\mathit{m}}_{\mathit{s}}^{\mathit{z}}$ of the staggered spin per particle ${\mathbf{m}}_{\mathit{s}}$ in the ground state of ${\mathit{H}}_1$ plus a symmetry-breaking term ${\mathit{H}}_{\mathit{B}}$ due to an infinitesimal staggered field B in the thermodynamic limit (TL). We find a value of √3 for the ratio r=m/${\mathit{m}}_0$, ${\mathit{m}}_0$ being the root-mean-square value of ${\mathit{m}}_{\mathit{s}}^{\mathit{z}}$ for B==0 in the TL. m and ${\mathit{m}}_0$ 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 ${\mathbf{m}}_{\mathit{s}}$ 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 ${\mathit{H}}_1$ 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 ${\mathit{N}}^{\mathrm{-}1}$, 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(${\mathit{N}}^0$); i.e., it is not extensive.