Euclidean Gibbs States of Quantum Lattice Systems

Albeverio, Sergio; Kondratiev, Yuri; Kozitsky, Yu.; Röckner, Michael

doi:10.1142/s0129055x02001545

Cited by 29 publications

(109 citation statements)

References 58 publications

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“…Now we can follow the line of reasoning of [2,7,32]. For every bounded function A(x Λ ) on IR d|Λ| we consider a bounded operator A 0 on H Λ defined as multiplication on a bounded function: 29) and for any t > 0 we consider the operator…”

Section: By Virtue Of (220) and (223) It Is Clear Thatmentioning

confidence: 99%

“…Following [7] we shall call this measure the Euclidean Gibbs Measure (EGM), which corresponds to our particular model (2.1)-(2.3) in this context. So, due to this construction the Theorem 2.1 for quantum Gibbs states (2.32)-(2.33) can be reformulated as follows: Theorem 2.2 For the system of quantum particles with Hamiltonian (2.1)-(2.3) there exists a sufficiently small mass m * such that for any 0 < m < m * the weak limit of measures (2.34)…”

Section: By Virtue Of (220) and (223) It Is Clear Thatmentioning

confidence: 99%

“…To prove the uniqueness of the limit measure in the Theorem 2.2 for non-periodic boundary conditions, we keep the DLR language (see [25]) and consider EGM, with some general boundary conditions ξ, from a class studied already in [6,7]. To this end we define the set of tempered configurations: we define the perturbed measure with non-zero boundary conditions by We also need the following auxiliary measureμ Λ (· | y Λ ) on Hβ ,Λ , which depends on some fixed trajectories y Λ ∈ Ωβ ,Λ :…”

Section: )mentioning

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: By Virtue Of (220) and (223) It Is Clear Thatmentioning

confidence: 99%

Section: By Virtue Of (220) and (223) It Is Clear Thatmentioning

confidence: 99%

Section: )mentioning

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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“…Lattice systems of such type are commonly viewed in quantum statistical physics as mathematical models of a crystalline substance (for more physical background see, e.g., [1,2,15,17]). A complete description of thermal equilibrium properties of quantum systems might be given in terms of their Gibbs states.…”

Section: The Modelmentioning

confidence: 99%

“…We follow here the Euclidean (or path space) approach, which remains so far the only method which allows to construct and study Gibbs states for infinite systems of quantum particles described by unbounded operators. This approach was first implemented to quantum lattice systems in [1]; for further developments see [2], [4], [8], [18], [19], [21]. Briefly speaking, we transform the problem of giving a proper meaning to a quantum Gibbs state ρ β into the problem of studying a certain Euclidean Gibbs measure µ on the 'temperature loop lattice' Ω := [C(S β )] L (cf.…”

Section: Introductionmentioning

confidence: 99%