Condensed Matter Physics, 2002, vol. 5, No. 4(32), p. 601-616, English

Authors: Yu.Kozitsky (Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland)

A model of quantum particles performing $D$-dimensional anharmonic oscillations around their equilibrium positions which form the $d$-dimensional simple cubic lattice $\mathbf{Z}^d$ 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 $\omega_n$, $n\in \mathbf{Z}$. We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero $\omega_n$ 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 $\omega_n = 0$) 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.

Key words: displacements, structural phase transition, critical point
PACS: 05.50.-d, 64.60.-i

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