Crystallization and Instabilities in Highly Anharmonic Crystals

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Abstract
A unified treatment of crystalline order and instabilities in highly anharmonic crystals at all temperatures is presented. This treatment is based on the study of singularities in the atomic-displacement correlation function or its Fourier transform (structure factor). As a result, it is rigorously shown that in the thermodynamic limit the mean-square fluctuations of the equilibrium position of a lattice particle are infinite in one and two dimensions at nonzero temperatures and in one dimension at zero temperature. This result, which is proved for an interacting many-body system without assuming the harmonic approximation, is obtained by using an exact Dyson equation for the displacement-response function. At finite temperatures, the demonstration of similar instabilities in a variety of condensed many-body systems of one and two dimensions is usually based on inequalities originally due to Bogoliubov. Since an analogous inequality can readily be extracted from the Dyson equation of the present approach, our method allows the extension of these results in anharmonic crystals to zero temperature. Finally, it is shown that additional dynamical information in this Dyson equality can be used to derive the relationship between elastic anomalies and sound absorption in the vicinity of critical points from the anomalous increase of the second derivative of the displacement-autocorrelation function.