Uniqueness for Gibbs measures of quantum lattices in small mass regime

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

doi:10.1016/s0246-0203(00)01057-8

Cited by 15 publications

(31 citation statements)

References 29 publications

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“…In recent years, there appeared a number of publications describing infuence of quantum effects on phase transitions in quantum anharmonic crystals, where the results were obtained by means of path integrals, see [3,5,8,9,45,48,50,56,63,82]. Their common point is a statement that the phase transition (understood in one or another way) is suppressed if the model parameters obey a sufficient condition (more or less explicitely formulated).…”

Section: Introduction and Setupmentioning

confidence: 99%

Phase Transitions and Quantum Stabilization in Quantum Anharmonic Crystals

2008

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A unified theory of phase transitions and quantum effects in quantum anharmonic crystals is presented. In its framework, the relationship between these two phenomena is analyzed. The theory is based on the representation of the model Gibbs states in terms of path measures (Euclidean Gibbs measures). It covers the case of crystals without translation invariance, as well as the case of asymmetric anharmonic potentials. The results obtained are compared with those known in the literature.

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Section: Introduction and Setupmentioning

confidence: 99%

Phase Transitions and Quantum Stabilization in Quantum Anharmonic Crystals

2008

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“…In fact, the uniqueness of the limiting Gibbs measures may be proved based on the more sophisticated methods. Here one has to mention that such a uniqueness was proved in [5] to hold for D = 1 and for the values of the particle mass from the interval (0, m * (β)), where the bound m * (β) tended to zero as β → +∞. In a very recent paper [7] it is stated that the uniqueness of the limiting Gibbs measures (again for D = 1 only) may be guaranteed by a condition similar to (19), which holds for m ∈ (0, m * ), where m * is independent of β.…”

Section: Discussionmentioning

confidence: 93%

Quantum Effects in an Anharmonic Crystal

Kozitsky¹

2002

Condens. Matter Phys.

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A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Z d is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ω n , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ω n remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ω n = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.

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“…No stability arguments were used. A deeper result of such stabilization turns out to be the uniqueness of the tempered Euclidean Gibbs measures, which was proven in [21,23].…”

Section: Quantum Effectsmentioning

confidence: 93%

“…A method of constructing Gibbs states of models like (1.1), based on functional integrals, was initiated in [11,12]. Now it is well elaborated, see [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The aim of this paper is to give a brief introduction into this method, accessible also to physicists.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Anharmonic Crystal in Functional Integral Approach

Kondratiev¹,

Kozitsky²,

Rockner³

2003

Condens. Matter Phys.

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A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals. The main aspects of this approach are described on an introductory level. Then a mechanism of the stabilization of this model by quantum effects is suggested. In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given. It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough. It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures.

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Uniqueness for Gibbs measures of quantum lattices in small mass regime

Cited by 15 publications

References 29 publications

Phase Transitions and Quantum Stabilization in Quantum Anharmonic Crystals

Phase Transitions and Quantum Stabilization in Quantum Anharmonic Crystals

Quantum Effects in an Anharmonic Crystal

Quantum Anharmonic Crystal in Functional Integral Approach

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