95-233 Borgs C., Kotecky R., Ueltschi D.

Low temperature phase diagrams for quantum perturbations of classical spin systems (113K, AMS-TeX) May 29, 95

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Abstract. We consider a quantum spin system with Hamiltonian $$ H=H^{(0)}+\lambda V, $$ where $H^{(0)}$ is diagonal in a basis $\ket s=\bigotimes_x\ket{s_x}$ which may be labeled by the configurations $s=\{s_x\}$ of a suitable classical spin system on $\Bbb Z^d$, $$ H^{(0)}\ket s=H^{(0)}(s)\ket s. $$ 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 {\it 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.

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