Mean-field calculation of the central peak due to the phonon-Ising dynamic crossover

Abstract

A self-consistent numerical calculation of the onset of the central mode in a phonon ${\phi }^4$ field theory, for $T>\mathrm{}{T}_c$, has been carried out. The central mode is attributed to crossover to Ising-like tunneling behavior, which is driven by indirect coupling to a collective excitation, corresponding to damped or overdamped second sound. Under the restrictive assumption that the central mode does not overlap the phonon sidebands, the self-consistency equation for the intensity parameter $\delta$ is found analogous to the BCS "gap" equation. Far from $T_c$, $\delta$ is identically zero. After onset above $T_c$, a 20-30% further change in coherence length is found sufficient for transfer of 80% of the phonon intensity into the central mode. The characteristic width of the (assumed Lorentzian) central mode is initially much broader than the phonon width, until about 80% intensity transfer occurs. Thereafter, critical narrowing is found to be anomalous. The formation of the central mode is characterized by a second short fluctuation length presumably controlling the surface energy between virtual domains. The (nonlocal) collective excitation presumably describes slow fluctuations into these virtual domains. The spectrum of the local two-phonon correlation function is found not to reflect the collective excitation but faster and more complicated zero-point motion which is not analyzed here. An extension of the calculation to resolve this spectrum and other possible structure in the central mode is not out of the question, but this will be left for future calculation.