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Journal of Physics A Mathematical and General

No-go theorem for quantum structural phase transitions

André F. Verbeure, Valentin Anatol'evich Zagrebnov

Published Sep 21, 1995• 34 citations
Publisher

Abstract

We prove that quantum fluctuations can suppress structural phase transitions. We give a rigorous proof for a one-component (R 1 ) quantum crystal with local double-well anharmonism under the condition that the masses of the atoms in the lattice sites of Z d (d>or=3) are light enough.
DOI
10.1088/0305-4470/28/18/029

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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 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.
In this paper we continue the investigation of the structural phase transition in a quantum anharmonic crystal. We calculate the influence of breaking the symmetry of the interaction potential on the behaviour of fluctuations and their critical exponents. More precisely, it is shown that when the low-momentum asymptotics of the phonon spectrum (i.e. the Fourier transform of the interaction matrix) have a quartic asymptotic in one direction, the critical exponents for the momentum and position fluctuations increase on the critical line. The quantum nature of the fluctuations in T = 0 is, however, preserved.