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 \ensuremath{\surd}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= \ensuremath{\surd}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 ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$, 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.
We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from
0 to
1, the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys.
18, 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin
S=1, our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.
The ground-state properties of the quasi-one-dimensional spin-
1/2 antiferromagnetic Heisenberg model is investigated by using a variational method. Spins on chains along the
x direction are antiferromagnetically coupled with exchange
J>0, while spins between chains in the
y direction are coupled either ferromagnetically (${J}^{\ensuremath{'}}<0$) or antiferromagnetically (${J}^{\ensuremath{'}}>0$). The staggered and the colinear antiferromagnetic magnetizations are computed and their dependence on the anisotropy parameter $\ensuremath{\lambda}=|{J}^{\ensuremath{'}}|/J$ is analyzed. It is found that an infinitesimal interchain coupling parameter is sufficient to stabilize a long-range order with either a staggered magnetization
ms (${J}^{\ensuremath{'}}>0$) or a colinear antiferromagnetic magnetization
mcaf (${J}^{\ensuremath{'}}<0$), both behaving as $\ensuremath{\simeq}{\ensuremath{\lambda}}^{1/2}$ for $\ensuremath{\lambda}\ensuremath{\rightarrow}0$.