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Quantum effects in an anharmonic crystal

Yuri V. Kozitsky

Published Jan 1, 2002• 0 citations
Publisher

Abstract

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

DOI
10.5488/cmp.5.4.601

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A model of quantum particles performing D -dimensional anharmonic oscillations \naround their equilibrium positions which form the d -dimensional \nsimple cubic lattice Zd is considered. The model undergoes a structural \nphase transition when the fluctuations of displacements of particles become \nmacroscopic. This phenomenon is described by susceptibilities depending \non Matsubara frequencies ωn , n ∈ Z . We prove two theorems \nconcerning the thermodynamic limits of these susceptibilities. The first theorem \nstates that the susceptibilities with nonzero ωn remain bounded at all \ntemperatures, which means that the macroscopic fluctuations in the model \nare always non-quantum. The second theorem gives a sufficient condition \nfor the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded \nat all temperatures. This condition involves the particle mass, the anharmonicity \nparameters and the interaction intensity. The physical meaning of \nthis result is that, for all D and all values of the temperature, strong quantum \neffects suppress critical points and the long range order. The proof \nis performed in the approach where the susceptibilities are represented as \nfunctional integrals. A brief description of the main features of this approach \nis delivered.
A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd 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 ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn 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 ωn = 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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An exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction (decreasing at large distances r as r−d−σ, where d is the space dimensionality and 0 < σ ≤ 2) is considered near to its zero-temperature critical point. The low-temperature behavior of the bulk specific heat capacity at the lower classical critical dimension (d = σ) is studied in different regions of the (T, λ)-phase diagram, where T is the temperature and λ is a parameter which switches on quantum fluctuations. From the results obtained one can see that when T → 0+ the specific heat capacity tends to zero in a different way in the regions: (a) renormalized classical region - as T raised to the second power and (b) quantum disordered region - exponentially.
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.
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.