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Kozitsky, Yuri

doi:10.1023/a:1007675606191

Cited by 12 publications

(1 citation statement)

References 11 publications

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“…, 0) with h ∈ R. In the remaining (ii) and (iii) cases, we have ν = 1 and hence h ∈ R. Thus, from now on we assume that h ∈ R in all cases. By (49), (41) and (47), we obtain:…”

Section: The Free Energymentioning

confidence: 99%

Phase Transitions and Quantum Effects in Anharmonic Crystals

Albeverio

Kondratiev

Kozitsky

et al. 2012

Int. J. Mod. Phys. B

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The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the nondifferentiability of the free energy density and on the appearance of ordering, are also discussed.

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“…, 0) with h ∈ R. In the remaining (ii) and (iii) cases, we have ν = 1 and hence h ∈ R. Thus, from now on we assume that h ∈ R in all cases. By (49), (41) and (47), we obtain:…”

Section: The Free Energymentioning

confidence: 99%

Phase Transitions and Quantum Effects in Anharmonic Crystals

Albeverio

Kondratiev

Kozitsky

et al. 2012

Int. J. Mod. Phys. B

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Gibbs States of a Lattice System of Quantum Anharmonic Oscillators

Kozitsky

2001

Noncommutative Structures in Mathematics and Physics

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Analyticity of the Gibbs State for a Quantum Anharmonic Crystal: No Order Parameter

Minlos

Pechersky

Zagrebnov

2002

Ann. Henri Poincaré

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We prove that for a ν-dimensional quantum crystal model of interacting anharmonic oscillators of mass m there exists m 0 such that in the light-mass domain 0 < m < m 0 the corresponding Gibbs state is analytic with respect to external field (conjugate to site displacements) for all temperatures T ≥ 0, i.e. including the ground state. This means that for the model with harmonic interaction and a symmetric double-well one-site potential, the light-mass quantum fluctuations suppress the symmetry breaking structural phase transition known in this model for ν ≥ 3 and m > M 0 ≥ m 0 , where M 0 is large enough. 922 Analyticity of the Gibbs State for a Quantum Anharmonic Crystal 923 phasize that similar to [15]-[18], our method does not solve the uniqueness problem in the DLR-sense in the light-mass domain at T = 0, since it does not prove the uniqueness of the Gibbs field of paths for this temperature, if the boundary trajectories do not have limited amplitudes.The main obstacle to making this conclusion is the noncompactness of the path ("spin") variables. However, since our Theorem 2 shows that in the light-mass domain m < m 0 there is no order parameter corresponding to the symmetry breaking structural phase transition that exists for m > M 0 , one could anticipate in this domain the uniqueness of the quantum Gibbs state for all temperatures including T = 0. For the proof in the case of the compact "spins" see [21]. Notice that this makes a striking difference between the quantum model and its classical analog corresponding formally to m → ∞ (or the Planck constant → 0): in the quantum case the suppression of any ordering is expected even at zero temperature, due to the tunneling microscopic quantum fluctuations for sufficiently light masses m < m 0 .

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Cited by 12 publications

References 11 publications

Phase Transitions and Quantum Effects in Anharmonic Crystals

Phase Transitions and Quantum Effects in Anharmonic Crystals

Gibbs States of a Lattice System of Quantum Anharmonic Oscillators

Analyticity of the Gibbs State for a Quantum Anharmonic Crystal: No Order Parameter

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