Gibbs State Uniqueness for Anharmonic Quantum Crystal with a Nonpolynomial Double-Well Potential

Abstract

We construct the Gibbs state for $\nu$-dimensional quantum crystal with site displacements from ${\mathbb{R}}^d$, $d\ge 1$, and with a one-site \textit{non-polynomial} double-well potential, which has \textit{harmonic} asymptotic growth at infinity. We prove the uniqueness of the corresponding {\it Euclidean Gibbs measure} (EGM) in the \textit{light-mass regime} for the crystal particles. The corresponding state is constructed via a cluster expansion technique for an arbitrary temperature $T\ge 0$. We show that for all $T\ge 0$ the Gibbs state (correlation functions) is analytic with respect to external field conjugated to displacements provided that the mass of particles $m$ is less than a certain value $m_{\ast }>0$. The high temperature regime is also discussed.

45 pages