## Low temperature phase diagram for quantum

perturbations of classical spin systemsC. Borgs, R. Kotecky and D. Ueltschi

Commun. Math. Phys.181, 409-446 (1996)We consider a quantum spin system with Hamiltonian H = H_0 + lambda V,

where H_0 is diagonal in a basis |s> = tensorial product |s_x> which may be labelled 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 aclassicalcontour 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.