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Physical review. B, Solid state

Dynamics of the Iongitudinal spin for the three-dimensional, spin-½ Ising model in a transverse field

T. N. Tommet, David L. Huber

Published Mar 1, 1975• 46 citations
Publisher

Abstract

The dynamics of the Iongitudinal (i.e., parrallel to the $z$ axis) spin fluctuations of the three-dimensional Ising model in a transverse field, $H=\ensuremath{-}\ensuremath{\Gamma}\ensuremath{\Sigma}{i}^{}{S}_{i}^{x}\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{\mathrm{ij}}^{}{J}_{\mathrm{ij}}{S}_{i}^{z}{S}_{j}^{z}$, is studied. An approximate expression for the relaxation shape function is given which gives a simple but realistic view of the dynamics for the entire temperature range ${T}_{c}<T<\ensuremath{\infty}$. Plots of the shape function are obtained and analyzed.

DOI
10.1103/physrevb.11.1971

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The dynamics of the Iongitudinal (i.e., parrallel to the zzz axis) spin fluctuations of the three-dimensional Ising model in a transverse field, H=\ensuremath−\ensuremathΓ\ensuremathΣiSix\ensuremath−(12)\ensuremathΣijJijSizSjzH=\ensuremath{-}\ensuremath{\Gamma}\ensuremath{\Sigma}{i}^{}{S}_{i}^{x}\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{\mathrm{ij}}^{}{J}_{\mathrm{ij}}{S}_{i}^{z}{S}_{j}^{z}H=\ensuremath−\ensuremathΓ\ensuremathΣiSix​\ensuremath−(21​)\ensuremathΣijJij​Siz​Sjz​, is studied. An approximate expression for the relaxation shape function is given which gives a simple but realistic view of the dynamics for the entire temperature range Tc<T<\ensuremath∞{T}_{c}<T<\ensuremath{\infty}Tc​<T<\ensuremath∞. Plots of the shape function are obtained and analyzed.
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 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.