A new form of the Mayer expansion in classical statistical mechanics

Brydges, David C.; Federbush, Paul

doi:10.1063/1.523586

Cited by 98 publications

(99 citation statements)

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“…Now we return to the proof of Lemma 3.1. Since 21) then taking into account dependence of b m and δ m on m (see (2.10)) and definition (3.21), we obtain by summing over trees η the estimate:…”

Section: Proposition 42 Letmentioning

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: Proposition 42 Letmentioning

confidence: 99%

“…By construction (3.9)-(3.10) the function G Λ,Xn 0 (t k , t k ′ ; (s) n−1 , 0) is a convex combination of "diagonalized" covariances (see [21], or [30,Sec.18.2]):…”

Section: Proposition 42 Letmentioning

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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“…, γ n )|, since the pair potential is not purely hard-core (and also not purely repulsive). We will rather make use of another well known "tree graph identity" originally proved in [3] (see also [4,1,17,16]). …”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We state the so called tree graph identity [3], [1], [4] by using the notations of [17] and [16]. We use the short notation I n = {1, 2, .…”

Section: Tree Graph Inequality For |φmentioning

confidence: 99%

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Abstract Polymer Models with General Pair Interactions

Procacci

2007

J Stat Phys

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polymer models with general pair interactions Aldo ProcacciDep. Matemática-ICEx, UFMG, CP 702 Belo Horizonte -MG, 30161-970 Brazil email: aldo@mat.ufmg.br Abstract A convergence criterion of cluster expansion is presented in the case of an abstract polymer system with general pair interactions (i.e. not necessarily hard core or repulsive). As a concrete example, the low temperature disordered phase of the BEG model with infinite range interactions, decaying polynomially as 1/r d+λ with λ > 0, is studied.

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