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Physical review. B, Condensed matter

Dynamics of the strong anisotropic three-dimensional Ising model in a transverse field

J. A. Plascak, F. C. Sá Barreto, Antonio S. T. Pires

Published Jan 1, 1983• 16 citations
Publisher

Abstract

The dynamics of the three-dimensional anisotropic spin-\textonehalf{} Ising model in a transverse field is studied above ${T}_{c}$ by a method in which the coupled-chain problem is reduced to that of a single chain in an effective field. The problem of a single chain is studied by means of Mori's continued-fraction expansion in a four-pole approximation. The moments are obtained by using the time-dependent correlation function calculated in the $c$-cyclic and in the improved $c$-cyclic approximations [J. Phys. A 13, 223 (1980)]. A comparison between these two approximations is done by calculating the critical temperature as function of the tunneling field. The one-dimensional correlation length is obtained, and it agrees well with the experimental results for the quasi-one-dimensional ferroelectric Cs${\mathrm{D}}_{2}$P${\mathrm{O}}_{4}$.

DOI
10.1103/physrevb.27.523

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Physical Review E•Oct 7, 2015•19 citations
The dynamics of the three-dimensional anisotropic spin-\textonehalf{} Ising model in a transverse field is studied above Tc{T}_{c}Tc​ by a method in which the coupled-chain problem is reduced to that of a single chain in an effective field. The problem of a single chain is studied by means of Mori's continued-fraction expansion in a four-pole approximation. The moments are obtained by using the time-dependent correlation function calculated in the ccc-cyclic and in the improved ccc-cyclic approximations [J. Phys. A 13, 223 (1980)]. A comparison between these two approximations is done by calculating the critical temperature as function of the tunneling field. The one-dimensional correlation length is obtained, and it agrees well with the experimental results for the quasi-one-dimensional ferroelectric CsD2{\mathrm{D}}_{2}D2​PO4{\mathrm{O}}_{4}O4​.
We derive the moments, at $T=\ensuremath{\infty}$, of the longitudinal spin-spin correlation function of the Ising model in a transverse field. The first 20 moments are obtained for the linear chain, the first 12 moments for the fcc lattice. We use a method of Nickel [J. Phys. C 7, 1719 (1914)] to construct the relaxation function and find excellent agreement with the exact results of Capel and Perk [Physica (Utrecht) 87A, 211 (1977)] in one dimension and extremely good convergence in three dimensions. Our method of analysis provides much better convergence than methods based on truncation of the continued-fraction representation.
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.
The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field hhh at T=0T=0T=0. This model corresponds to the XYX YXY model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of hhh, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field h=hc=1h={h}_{c}=1h=hc​=1, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For h<1,h>1h<1,h>1h<1,h>1 the results already obtained for the XY\mathrm{XY}XY model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at T=0T=0T=0.
We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-1/2 transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.