An Ergodic Description of Ground States

Abstract

Given a translation-invariant Hamiltonian $H$, a ground state on the lattice $\mathbb {Z}^d$ is a configuration whose energy, calculated with respect to $H$, cannot be lowered by altering its states on a finite number of sites. The set formed by these configurations is translation-invariant. Given an observable $\Psi$ defined on the space of configurations, a minimizing measure is a translation-invariant probability which minimizes the average of $\Psi$. If $\Psi _0$ is the mean contribution of all interactions to the site $0$, we show that any configuration of the support of a minimizing measure is necessarily a ground state.