Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States

Abstract

A class of general spin 1/2 lattice models on hyper-cubic lattice $Z^d$, whose Hamiltonians are sums of two functions depending on the Pauli matrices $S^1$, $S^2$ and $S^3$, respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components $x$, $z$ simultaneously for large values of the parameter $\alpha$ playing the role of the inverse temperature. It is shown that the ferromagnetic order in $x$ direction exists for all dimensions $d\geq 1$ for a wide class of considered models (a proof is remarkably simple).