Euclidean Gibbs states of interacting quantum anharmonic oscillators

Abstract

A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting $\nu$-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set $\mathbb{L}\subset {\mathbb{R}}^d$, possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures -- the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures ${\mathcal{G}}^{\mathrm{t}}$ is non-void and compact; (b) every $\mu \in {\mathcal{G}}^{\mathrm{t}}$ obeys an exponential integrability estimate, the same for the whole set ${\mathcal{G}}^{\mathrm{t}}$; (c) every $\mu \in {\mathcal{G}}^{\mathrm{t}}$ has a Lebowitz-Presutti type support; (d) ${\mathcal{G}}^{\mathrm{t}}$ is a singleton at high temperatures. In the case of attractive interaction and $\nu =1$ we prove that $|{\mathcal{G}}^{\mathrm{t}}|>1$ at low temperatures. The uniqueness of Gibbs measures due to quantum effects and at a nonzero external field are also proven in this case. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed. The mathematical result of the paper is a complete description of the set ${\mathcal{G}}^{\mathrm{t}}$, which refines and extends the results known for models of this type.

60 pages

References in corpus (5)

• Phase Transitions on Nonamenable Graphs
• The ground state of a general electron-phonon Hamiltonian is a spin singlet
• The anharmonic electron-phonon problem
• KMS, etc
• Euclidean Gibbs Measures of Quantum Anharmonic Crystals

Cited by in corpus (8)

• Gibbs States on Random Configurations
• A quantum Mermin--Wagner theorem for quantum rotators on two--dimensional graphs
• Gibbs measures of disordered lattice systems with unbounded spins
• Phase Transitions and Quantum Stabilization in Quantum Anharmonic Crystals
• Phase transitions and quantum effects in anharmonic crystals
• Gibbs Measures on Marked Configuration Spaces: Existence and Uniqueness
• Gibbs Fields: Uniqueness and Decay of Correlations. Revisiting Dobrushin and Pechersky
• Phase Transitions in Continuum Ferromagnets with Unbounded Spins