Borgs C., Kotecky R., Ueltschi D.
Low temperature phase diagrams for quantum perturbations
of classical spin systems
(113K, AMS-TeX)
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.