Quantum spin XXZ-type models with Gibbsian ground states and double long-range order

A wide class of spin j  , j∈12Z+, lattice models on hyper-cubic lattice ZdZd are introduced, whose Hamiltonians have Gibbsian eigen (ground) states with the zero eigenvalue and lro (long-range order) in the third spin component for a special choice of a classical potential energy characterising this state. It is shown, also, that the ferromagnetic lro in the first spin component always exists for all dimensions for a wide subclass of considered models. A proof of this fact is remarkably simple. Among the systems introduced, there are systems in which U(1)U(1) symmetry is broken and which are characterized by many ground states. The Hamiltonians considered are simply related with generators of stationary Markovian processes.