Exactly solvable d-dimensional model of a structural phase transition

Плакида, Н. М.; Tonchev, N. S.

doi:10.1007/bf01017907

Cited by 20 publications

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“…In the general case this follows from [15]; see also [47]. On the other hand, in [48] and [54]- [58] it is shown that there is a non-zero threshold mass m * such that the phase transition in the displacement quantum spherical model is suppressed by the 'strong quantumness' as soon as m < m * .…”

Section: Introductionmentioning

confidence: 81%

“…In general case this follows from [15], see also [47]. On the other hand, in [48] and [54,58] it is shown that there is a non-zero threshold mass m * such that the phase transition in the displacement quantum spherical model is suppressed by the "strong quantumness" as soon as m < m * . The aim of the present paper is study the case of the quadratic growth, s = 1, of the double-well anharmonic one-site potential W (1.5).…”

Section: Introductionmentioning

confidence: 96%

“…It is easy to see that for a < bδ the minimum of the potential (1.5) is not unique and the system may undergo a displacement phase transition. For the case of the so-called self-consistent phonon approximation [1], or for the displacement quantum spherical model, this was proved in [48] for (1.3) and in [54]- [58] for (1.5). In the general case this follows from [15]; see also [47].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

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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“…In particular, this indicates that in the quantum case the bound m > M in the proofs [18][19][20]25] has not only a technical meaning! Justifications on the physical level [28,30] and calculations within exactly soluble models [1,31,32], show that quantum fluctuations may suppress the long-range order of structural displacements for all temperatures including θ = 0. A mathematical proof of this phenomenon for a one-component quantum ferroelectric model was obtained in [8].…”

Section: Introductionmentioning

confidence: 99%

A Quantum Crystal Model in the Light-Mass Limit: Gibbs States

2000

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Ground and temperature quantum Gibbs states are constructed for a ferroelectric anharmonic quantum oscillator model with small masses. It is shown that they possess mixing properties. The construction relies on the Feynman-Kac-Nelson representation of the conditional reduced density matrices and on the cluster expansions for the corresponding Gibbs field of trajectories. 1004 6. Cluster Expansions of Measures M 2T,Λ and M

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Models of a structural phase transition with general anharmonicity and disorder

Hemmen

Zagrebnov

1988

J Stat Phys

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Exactly solvable d-dimensional model of a structural phase transition

Cited by 20 publications

References 4 publications

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

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

A Quantum Crystal Model in the Light-Mass Limit: Gibbs States

Models of a structural phase transition with general anharmonicity and disorder

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