Low Temperature Phase Diagrams for Quantum Perturbations of Classical Spin Systems

We consider a quantum spin system with Hamiltonian H = H + λV, where H(0) is diagonal in a basis |s⟩ = ⊗ x |sx⟩ which may be labeled by the configurations s = {sx} of a suitable classical spin system on Zd, H |s⟩ = H(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 λ 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.