Absence of Critical Points for a Class of Quantum Hierarchical Models

Annales De l'Institut Henri Poincare (B) Probability and Statistics

Kondratiev

Kozitsky

et al. 2001

A model of interacting identical quantum particles performing one-dimensional anharmonic oscillations around their unstable equilibrium positions, which form the d-dimensional simple cubic lattice Z d , is considered. For this model it is proved that for every fixed value of the temperature β −1 there exists a positive m * (β) such that for the values of the physical mass of the particle m ∈ (0, m * (β)), the set of tempered Gibbs measures consists of exactly one element. © 2001 Éditions scientifiques et médicales Elsevier SAS AMS classification: 60B05, 82B10RÉSUMÉ. -On considère sur le réseau Z d un modèle de particules quantiques en interaction soumises à oscillations unidimensionnelles anharmoniques autour de leur position d'équilibre instable. Pour ce modèle on montre que pour chaque valeur fixée de la température β −1 il existe un réel m * (β) > 0 tel que pour les valeurs de la masse m ∈ [0, m * (β)], l'ensemble des état de Gibbs tempérés à un seul élément.

Section: Introductionsupporting

confidence: 55%

Uniqueness for Gibbs measures of quantum lattices in small mass regime

Annales De l'Institut Henri Poincare (B) Probability and Statistics

Kondratiev

Kozitsky

et al. 2001

“…Let E n , n ∈ N 0 be its eigenvalues and ∆ = min n∈N (E n − E n−1 ). In [4] we proved that if m∆ 2 > 1, then u 0 < 1 and henceû n → 0 for all β. In what follows, the critical point of the model exists if a < 0 and the parameters m(|a|/b) 2 , 1/b are big enough; such a point does not exist if 'the quantum rigidity' m∆ 2 (see [7]) is greater than 1.…”

Section: Kmentioning

confidence: 84%

“…In what follows, the critical point of the model exists if a < 0 and the parameters m(|a|/b) 2 , 1/b are big enough; such a point does not exist if 'the quantum rigidity' m∆ 2 (see [7]) is greater than 1. By Lemma 1.1 of [4], m∆ 2 ∼ m −1/3 C, C > 0 as m → 0, which means that small values of the mass prevent the system from criticality.…”

Section: Kmentioning

confidence: 94%

“…A theorem describing the critical point convergence was announced in [3]. In [4] we have shown that the critical point of the model (1.1) can be suppressed by strong quantum effects, which take place, in particular, when the mass m is less than a certain bound m * 2 . In the present paper we give a complete proof of the critical point convergence, which appears for sufficiently large values of the mass (see the discussion at the very end of this introduction).…”

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

A Hierarchical Model of Quantum Anharmonic Oscillators: Critical Point Convergence

Kondratiev

Kozak

et al. 2004

Commun. Math. Phys.

A hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the Euclidean approach, in which the local Gibbs states are constructed as measures on infinite dimensional spaces. The local states restricted to the subalgebra generated by fluctuations of displacements of particles are in the center of the study. They are described by means of the corresponding temperature Green (Matsubara) functions. The result of the paper is a theorem, which describes the critical point convergence of such Matsubara functions in the thermodynamic limit.

“…[BC]). Let us mention in this connection an e ect of suppression of the critical behaviour in quantum anharmonic cristalls which was obtained rigorously in the recent papers [VZ2], [AKK1], [AKK2].…”

Section: Examplesmentioning

confidence: 99%

Uniqueness of Gibbs states for quantum lattice systems

Probab Theory Relat Fields

Kondratiev²,

Röckner

et al. 1997

Self Cite

We prove uniqueness of Euclidean Gibbs states for certain quantum lattice systems with unbounded spins. We use Dobrushin's uniqueness criterion. The necessary estimates for the Vasershtein distance between the corresponding one-point conditional distributions with boundary conditions di ering only at one side, are obtained by proving a Log-Sobolev inequality on the inÿnite dimensional single spin (= loop) spaces. Some important classes of concrete examples to which all this applies are discussed.