Defect-induced condensation and central peak at structural transitions

We investigate how the statics and dynamics of distortive phase transitions are influenced by defects which locally increase the transition temperature. In contrast to the local condensation at a single defect, a finite concentration n of randomly distributed defects induces a real phase transition at the temperature T_c(n). The phonon response function is calculated analytically in single-site approximation. For T > T_c(n) and low phonon damping we find a soft-phonon impurity band below the continuum, which eventually becomes overdamped when T T_c(n) and produces a dynamical central peak in the phonon-phonon correlation function. In the case of stronger damping and low defect concentration, the impurity band is masked and only appears as a narrow central peak very close to T_c(n). We develop a method which allows the computation of the order parameter and the correlation function in random systems. For T << ( T_c(n), the dynamical central peak disappears in the longitudinal correlation function, while it persists in the transverse components for rotational symmetry. The effect of cubic terms breaking the continuous rotational invariance is also investigated. In any case, the finite order parameter below T_c(n) produces a static central peak in the dynamical structure factor, which has a finite width in momentum space.