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Ferroelectrics

Mode-coupling theory for the central peak in the ising model in a transverse field

Peter Prelovšek, Igor Sega

Published Jan 1, 1980• 3 citations
Publisher

Abstract

The dynamics of longitudinal spin fluctuations in the S = 1/2 Ising model in a transverse field is studied. Exploiting the mode-coupling approximation for the relaxation function we show that the width of the central peak which appears just above Tc is determined by the nonlinear coupling of fluctuations to the hydrodynamic energy-density mode. The results for the central peak are compared to those obtained within the usual three-pole approximation.

DOI
10.1080/00150198008008159

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The dynamics of longitudinal spin fluctuations in the S = 1/2 Ising model in a transverse field is studied. Exploiting the mode-coupling approximation for the relaxation function we show that the width of the central peak which appears just above Tc is determined by the nonlinear coupling of fluctuations to the hydrodynamic energy-density mode. The results for the central peak are compared to those obtained within the usual three-pole approximation.
Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.
Abstract The Ising model in a transverse field is treated by means of Green function technique to calculate the dynamic structure factor above and below Tc. The soft mode is overdamped in the disordered phase and near Tc, while it is nearly undamped at low temperatures. The coupling between the transverse soft mode and the relaxing longitudinal mode produces a central mode in the dynamic structure factor. The temperature dependence of the dynamic structure factor for zero wave vector is calculated numerically.
The molecular-dynamics technique is applied to the study of the classical ($S=\ensuremath{\infty}$) Ising model in a transverse field on a simple-cubic lattice. The results for the longitudinal- and transverse-relaxation shape function are presented for a set of temperatures in the disordered phase at one value of the transverse field. The longitudinal spectra at $T=\ensuremath{\infty}$ and in the critical regime are analyzed via the Mori's continued-fraction expansion and compared with various approximate results within this approach. The relation of the observed central mode to a domain-wall motion is also discussed. However, due to heavy-damping phenomena no well-defined propagating modes are observed in the critical region. Finally, transverse correlation functions are tested against their diffusive character.