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Physical Review Letters

Some Exact Results for Multiorientational "Ising-Type" Dipoles in Transverse Fields

Michael W. Klein, Zhong He Wang

Published Sep 15, 1986• 8 citations
Publisher

Abstract

We consider a system of ${2}^{n}$ orientational interacting tunneling dipoles in $n$ transverse near-neighbor-well tunneling fields $^{\ensuremath{\alpha}}\ensuremath{\Delta}$ ($1>~\ensuremath{\alpha}>~n$). We show that the partition function for the system can be expressed as a product of $n$ Ising-model partition functions in transverse fields. The results are valid in one, two, and three dimensions as well as for random $\ensuremath{\Delta}$ s and random general interaction potentials.

DOI
10.1103/physrevlett.57.1355

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We consider a system of 2n{2}^{n}2n orientational interacting tunneling dipoles in nnn transverse near-neighbor-well tunneling fields \ensuremathα\ensuremathΔ^{\ensuremath{\alpha}}\ensuremath{\Delta}\ensuremathα\ensuremathΔ (1> \ensuremathα> n1>~\ensuremath{\alpha}>~n1> \ensuremathα> n). We show that the partition function for the system can be expressed as a product of nnn Ising-model partition functions in transverse fields. The results are valid in one, two, and three dimensions as well as for random \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ s and random general interaction potentials.
We consider an infinite chain of four-orientational hindered dipoles with near-neighbor interaction J. The dipoles are allowed to undergo tunneling into the near-neighbor wells (90\ifmmode^\circ\else\textdegree\fi{} tunneling) on the same site. We show that the partition function ZD{Z}_{D}ZD​(J) of the dipoles is ZD{Z}_{D}ZD​(J)=ZS2{Z}_{S}^{2}ZS2​(J/2), where ZS{Z}_{S}ZS​ is the partition function for a chain of Ising-model spins in a transverse field similar to that treated by Pfeuty. The dipoles act like a pair of independently embedded Ising-spin chains in a transverse field.
The partition function for the Ising model in a transverse field can be expressed as a sum over the usual Ising states by evaluation of the diagonal matrix elements between Ising states with the use of the cumulant expansion. The cumulant sum is evaluated to all orders in \ensuremath{\beta} and second order in \ensuremath{\Gamma}, the transverse field. The effective Hamiltonian is then employed in Monte Carlo simulation to evaluate the average energy and magnetization in the longitudinal and transverse directions. Comparison with exact results on a one-dimensional chain shows that the method works quite well. Monte Carlo results for the average energy, specific heat, and the transverse and longitudinal components of the magnetization for the transverse Ising model on the simple cubic lattice are presented.
The transverse-field Ising models with random exchange interactions in finite dimensions are investigated by means of a real-space renormalization-group method. The scheme yields the exact values of the critical point and critical exponent $\ensuremath{\nu}$ in one dimension and some previous results in the case of random ferromagnetic interactions are reproduced in two and three dimensions. We apply the scheme to spin glasses in transverse fields in two and three dimensions, which have not been analyzed very extensively. The phase diagrams and the critical exponent $\ensuremath{\nu}$ are obtained and evidence for the existence of an infinite-randomness fixed point in these models is found.
We consider the random transverse-field Ising model in d=3d=3d=3 dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL,L1/lnL, L1/lnL,L being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.
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​.
Dynamics and statistical mechanics of kinks in one-dimensional classical Ising model in a transverse field is studied in a dilute kink-gas approximation along the line with the theory of Krumhansl and Schrieffer. The energy of a single kink is written in a simple analytic form by approximating an exact one-kink solution. Kinks in the system are then treated as gas of quasi-particles. It is shown that the partition function so obtained is coincident with that of the result obtained previously by using the transfermatrix method. The dynamical structure of the system is also calculated to show the existence of central peak due to kink excitations.