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

TL;DR

This paper constructs and proves the uniqueness of the Gibbs state for a quantum crystal with a non-polynomial double-well potential, using cluster expansion techniques across all temperatures.

Contribution

It introduces a method to establish the uniqueness of Euclidean Gibbs measures for quantum crystals with non-polynomial potentials in the light-mass regime.

Findings

Gibbs state constructed for all temperatures using cluster expansion.

Gibbs state is analytic in external field for particle mass below a threshold.

Uniqueness of Gibbs measure proven in the light-mass regime.

Abstract

We construct the Gibbs state for $\nu$-dimensional quantum crystal with site displacements from $\R^d$, $d\geq 1$, 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 $T\geq 0$. We show that for all $T\geq 0$ 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 $m_* >0$. The high temperature regime is also discussed.