Quantum Anharmonic Crystal in Functional Integral Approach

Abstract

A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals. The main aspects of this approach are described on an introductory level. Then a mechanism of the stabilization of this model by quantum effects is suggested. In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given. It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough. It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures.

Decay of correlations in N-vector ferromagnetic quantum models with long-range interactions

“…describes an isolated anharmonic oscillator of mass m and momentum p l ; the part H har l describes an N -dimensional harmonic oscillator of rigidity a, see [2,[10][11][12][13][14][15][16] for more details on this model. Note that here we use units such that = 1.…”

“…In the Euclidean approach [11,12,14,15], such local states are constructed by means of probability measures on spaces of β-periodic continuous functions (temperature loops). Let C(R β → R N ) stand for the set of all continuous functions on the circle R β ∼ [0, β] with Lebesgue measure dτ and distance…”

Equilibrium dynamics and phase transitions in quantum anharmonic crystals

“…where ZA is as in (9). It can be viewed as the state of thermal equilibrium of the considered portion of oscillators at temperature T = l/fe/3.…”

“…In the Euclidean approach which we follow in this note, the description of thermodynamic phases of the model (5) existing at a given /3 is made by constructing the set of tempered Euclidean Gibbs measures ^l, see [2,9,14]. By definition, each ^u G ^o is a probability measure on the space of tempered configurations S', which solves the socalled equilibrium (another name Dobrushin-Lanford-Ruelle) equation^, formulated with the aid of the Hamiltonians (8).…”