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Physical review. B, Condensed matter

Memory functions for the one-dimensional Ising model in a transverse field

J. Oitmaa, I. Liubarsky, M. Aydin

Published Sep 1, 1989• 5 citations
Publisher

Abstract

We have computed the sequence of memory functions ${K}_{n}$(t) and their Laplace transforms, for the one-dimensional Ising model in a transverse field at T=\ensuremath{\infty}. The form of these functions shows explicitly why the Mori continued-fraction approach, with traditional truncation schemes, fails for this model.

DOI
10.1103/physrevb.40.5201

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We have computed the sequence of memory functions Kn{K}_{n}Kn​(t) and their Laplace transforms, for the one-dimensional Ising model in a transverse field at T=\ensuremath{\infty}. The form of these functions shows explicitly why the Mori continued-fraction approach, with traditional truncation schemes, fails for this model.
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.
Dynamical correlation functions for the transverse (along the direction of the field) spin and energy densities are calculated for the one-dimensional spin-\textonehalf{} Ising model with a transverse field $H=\ensuremath{-}\ensuremath{\Gamma}{\ensuremath{\Sigma}}_{i}{S}_{i}^{x}\ensuremath{-}J{\ensuremath{\Sigma}}_{i}{S}_{i}^{z}{S}_{i+1}^{z}$. Explicit results are obtained for $〈{{S}^{x}(q, \ensuremath{\omega}){S}^{x}(\ensuremath{-}q)}〉$, $〈{E(q, \ensuremath{\omega}), E(\ensuremath{-}q)}〉$, and their Fourier transforms in the limits $T=\ensuremath{\infty}$ and T=0T=0T=0 and for the autocorrelation functions 〈Six,Six(t)〉〈{{S}_{i}^{x}, {S}_{i}^{x}(t)}〉〈Six​,Six​(t)〉 and 〈Ei,Ei(t)〉〈{{E}_{i}, {E}_{i}(t)}〉〈Ei​,Ei​(t)〉 in the same limits. The results are discussed with special attention given to the coupling of the transverse spin component and energy densities.