, Volume 181, Issue 2, pp 409-446

Low temperature phase diagrams for quantum perturbations of classical spin systems

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Abstract

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH (0) is diagonal in a basis ∣s〉=⊗ x s x 〉 which may be labeled by the configurationss={sx} of a suitable classical spin system on ℤ d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH (0)(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+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 HamiltonianH (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.

Partly supported by the grants GAČR 202/93/0449 and GAUK 376.
Communicated by Ya.G. Sinai