Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian H=H-(0)+lambda V, where H-(0) is diagonal in a basis /s]=X(x)/s(x)] which may be labeled by the configurations s={s(x)} of a suitable classical spin system on Z(d), H-(0)/s]=H-(0)(s)/s]. We assume that H-(0)(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitations, while V is a finite range or exponentially decaying quantum perturbation. Mapping the d dimensional quantum system onto a classical contour system on a d+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical Hamiltonian H-(0), provided lambda is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.