Comptes Rendus
Gibbs states of a quantum crystal: uniqueness by small particle mass
Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 693-698.

A model of interacting quantum particles performing one-dimensional anharmonic oscillations around their unstable equilibrium positions, which form the lattice d , is considered. For this model, two statements describing its equilibrium properties are given. The first theorem states that there exists m * >0 such that for all values of the particle mass m<m * , the set of tempered Euclidean Gibbs measures consists of exactly one element at all values of the temperature β−1. This settles a problem that was open for a long time and is an essential improvement of a similar result proved before by the same authors [1] where the boundary m * depended on β in such a way that m * (β)0 for β→+∞. The second theorem states that the two-point correlation function has an exponential decay if m<m * .

On considère un modèle de particules quantiques en intéraction effectuant des oscillations anharmoniques uni-dimensionelles autour de leur positions d'équilibre sur le réseau d . Pour ce modèle, nous énonçons deux résultats décrivant ses propriétés d'équilibre. Le premier théorème affirme l'existence de m * >0 tel que pour toutes les valeurs de la masse m de la particule inférieures à m * , l'ensemble des mesures euclidiennes tempérées de Gibbs consiste en un seul élément, à toute température β−1. Cela résoud un problème qui est resté ouvert pour longtemps et améliore essentiellement un résultat analogue obtenu par les mêmes auteurs, lorsque m * dépendait de β de sorte que m * (β)0 si β→+∞. Le deuxième théorème dit que la fonction de corrélation a une décroissance exponentielle si m<m * .

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DOI: 10.1016/S1631-073X(02)02545-1
Sergio Albeverio 1, 2, 3; Yuri Kondratiev 4, 2, 5; Yuri Kozitsky 6; Michael Röckner 4, 2

1 Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
2 Forschungszentrum BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany
3 CERFIM, Locarno and USI, Switzerland
4 Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
5 Institute of Mathematics, Kiev, Ukraine
6 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland
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     title = {Gibbs states of a quantum crystal: uniqueness by small particle mass},
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Sergio Albeverio; Yuri Kondratiev; Yuri Kozitsky; Michael Röckner. Gibbs states of a quantum crystal: uniqueness by small particle mass. Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 693-698. doi : 10.1016/S1631-073X(02)02545-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02545-1/

[1] S. Albeverio; Yu. Kondratiev; Yu. Kozitsky; M. Röckner Uniqueness for Gibbs measures of quntum lattices for small mass regime, Ann. Inst. H. Poincaré, Probab. Statist., Volume 37 (2001) no. 1, pp. 43-69

[2] S. Albeverio, Yu. Kondratiev, Yu. Kozitsky, M. Röckner, Euclidean Gibbs states for quantum lattice systems, Preprint BiBoS, Bielefeld, 2001, to appear in Rev. Math. Phys

[3] S. Albeverio; Yu. Kondratiev; Yu. Kozitsky Suppression of critical fluctuations by strong quantum effects in quantum lattice systems, Comm. Math. Phys., Volume 194 (1998), pp. 493-512

[4] S. Albeverio, Yu. Kondratiev, T. Pasurek, M. Röckner, A priori estimates and existence for Euclidean Gibbs measures, Preprint BiBoS Nr. 02-06-089, Bielefeld, 2002

[5] V.S. Barbulyak; Yu.G. Kondratiev The quasiclassical limit for the Schrödinger operator and phase transitions in quantum statistical physics, Func. Anal. Appl., Volume 26 (1992) no. 2, pp. 61-64

[6] H.-O. Georgii Gibbs Measures and Phase Transitions, De Gruyter, Berlin, 1988

[7] J.L. Lebowitz; E. Presutti Statistical mechanics of unbounded spins, Comm. Math. Phys., Volume 50 (1976), pp. 195-218

[8] R.A. Minlos; A. Verbeure; V.A. Zagrebnov A quantum crystal model in the light-mass limit: Gibbs states, Rev. Math. Phys., Volume 12 (2000), pp. 981-1032

[9] T. Schneider; H. Beck; E. Stoll Quantum effects in an n-component vector model for structural phase transitions, Phys. Rev. B, Volume 13 (1976), pp. 1123-1130

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