Uniqueness of Gibbs states for quantum lattice systems

Albeverio, Sergio; Kondratiev, Yuri; Röckner, Michael; Tsikalenko, T. V.

doi:10.1007/s004400050107

Cited by 34 publications

(71 citation statements)

References 30 publications

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“…Then, to complete the proof of the uniqueness of the EGM, in the next section we consider also a general boundary conditions generated by some fixed configurationω from the setΩ tβ , which includes so-called tempered configurations, see [6,13,14] and [46]. Let L(H Λ ) be algebra of bounded operators on H Λ .…”

Section: Description Of the System And Main Resultsmentioning

confidence: 99%

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Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential

Rebenko¹,

Zagrebnov²

2006

J. Stat. Mech.

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We construct the Gibbs state for ν-dimensional quantum crystal with site displacements from IR d , d ≥ 1, and with a one-site non-polynomial double-well potential, which has harmonic asymptotic growth at infinity. We prove the uniqueness of the corresponding Euclidean Gibbs measure (EGM) in the lightmass regime for the crystal particles. The corresponding state is constructed via a cluster expansion technique for an arbitrary temperature T ≥ 0. We show that for all T ≥ 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.Keywords : quantum crystal model, Gibbs state, Euclidean Gibbs measures, quantum fluctuations, light-mass regime, cluster expansions. Mathematics Subject Classification : 60H30, 82B31 IntroductionIt is generally excepted that for investigation of different physical phenomena in quantum crystals one can consider an infinite system of interacting anharmonic oscillators, which are situated in the sites of ν-dimensional lattice Z Z ν . The heuristic Hamiltonian of such a system (in the case of 2-body interaction) has the following form:where m is the mass of particles, the operator ∆ j corresponds the kinetic energy of the system and, in fact, is d-dimensional Laplace operator in the one-particle Hilbert space, where dq is the Lebesgue measure onν is displacement of a particle from its position in the site j ∈ Z Z ν . For general case d ≤ ν ≥ 1. But the most of results, which where obtained earlier are for d = 1. The particles are confined near their sites by potential W (q j ). A harmonic one-site potential W harm (q j ) = 1 2 aq 2 j , a > 0 together with a harmonic two-particle interaction Q(·, ·) define a well-known harmonic crystal model (1.1). To produce a model describing a (ferroelectric) structural phase transition one usually takes for W (·) a doublewell anharmonic potential, keeping Q(·, ·) harmonic (see e.g. [1,19,24]). For example,or semibounded from below polynomials of higher degree with more than two equal minima. Here q 2 ≡ q · q is the scalar square of displacement vector q ∈ IR d . The proof of existence of phase transition in such kind of systems for harmonic interaction Q(q i , q j ) were obtained in [12,15,26,34,40,47,53] for the case2) whereQ = (Q ij ) i,j∈Z Z ν is a matrix of non-negative harmonic-force constants. The term h · q j = d α=1 h α q α j break the symmetry in the direction of the 1 external field h. The non-uniqueness of Gibbs states for h = 0 is proved when the mass of the particles m is sufficiently large and the temperature T is sufficiently low, or β = (kT ) −1 is sufficiently large. From the other side, another interesting phenomenon, the suppression of the long-range order by strong quantum fluctuations in such systems was experimentally observed (see, e.g. [55]) and was discussing long time ago from the physical point of view, see [51], or the books [1,19]. A rigorous stu...

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Section: Description Of the System And Main Resultsmentioning

confidence: 99%

“…in [13,14]. But there exist an independent question of construction of high temperature expansions for quantum state (2.5) and of the proof of its convergence in thermodynamic limit.…”

Section: Convergence Of Cluster Expansions and Gibbs State Uniquenessmentioning

confidence: 99%

Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential

Rebenko¹,

Zagrebnov²

2006

J. Stat. Mech.

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“…Nous décrivons les propriétés d'équilibre de ces systèmes quantiques au moyen desétats de Gibbs qui sont donnés par des mesures de Gibbs ( euclidiennes ) µ β associées avec H et une température inverse β > 0 [1,2]. La définition rigoureuse de µ β est comme suit.…”

Section: Version Française Abrégéeunclassified

“…We will take the Euclidean (i.e., path space) approach, see e.g. [1,2] and the references therein. Therewith the Euclidean Gibbs measures µ β associated with the lattice system (1) at the inverse temperature β > 0 are rigorously defined as follows:…”

Section: Quantum Crystals and Euclidean Gibbs Measuresmentioning

confidence: 99%

Gibbs states on loop lattices: existence and a priori estimates

Albeverio

Kondratiev

Pasurek

et al. 2001

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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Abstract. We prove existence and uniform a priori estimates for Euclidean Gibbs states of quantum lattice systems with unbounded spins. These results substantially extend all previous existence results. Detailed proofs are contained in [3].Etats de Gibbs sur des réseaux de lacets: existence et estimationsà priori Résumé. Des estimationsà priori et l'existence desétats de Gibbs pour des systèmes quantiques sur réseaux avec spins non bornés sont données. Ces résultatś etendent substanciellement tous les résultats d'existence précédents. Les preuves détail-lées sont contenues dans [3].

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“…(iii) The uniqueness problem for µ ∈ G t could be treated in the particular case of R = 2 by means of the another renown criterion of Dobrushin, see Theorem 4 in [13] and its applications to the quantum lattice systems in [5], [6]. However, this criterion is typically not applicable to the pair interactions W growing fastly than quadratic as considered here.…”

Section: Corollary 55 (Lebowitz-presutti Support)mentioning

confidence: 99%