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arXiv:cond-mat/9904036 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Exact correlation functions of Bethe lattice spin models in external magnetic fields

Chin‐Kun Hu, N. Sh. Izmailian

Published Aug 1, 1998• 47 citations
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Abstract

We develop a transfer matrix method to compute exactly the spin-spin correlation functions $〈{s}_{0}{s}_{n}〉$ of Bethe lattice spin models in the external magnetic field $h$ and for any temperature $T.$ We first compute $〈{s}_{0}{s}_{n}〉$ for the most general spin-$S$ Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-$S$ Ising model includes the spin-$\frac{1}{2}$ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and $T$ approaches the critical temperature ${T}_{c}.$ Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.

DOI
10.1103/physreve.58.1644

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We develop a transfer matrix method to compute exactly the spin-spin correlation functions 〈s0sn〉〈{s}_{0}{s}_{n}〉〈s0​sn​〉 of Bethe lattice spin models in the external magnetic field hhh and for any temperature T.T.T. We first compute 〈s0sn〉〈{s}_{0}{s}_{n}〉〈s0​sn​〉 for the most general spin-SSS Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-SSS Ising model includes the spin-12\frac{1}{2}21​ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length \ensuremathξ(T,h)\ensuremath{\xi}(T,h)\ensuremathξ(T,h) for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as h\ensuremath→0\stackrel{\ensuremath{\rightarrow}}{h}0h\ensuremath→​0 and TTT approaches the critical temperature Tc.{T}_{c}.Tc​. Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.
Multisite interaction spin-S models in an external magnetic field are studied recursively on the Bethe-like lattices. The transfer-matrix method is extended to calculate exactly the two-spin correlation functions. The exact expressions for the correlation length and magnetic susceptibility are derived for spin-1/2 models. The singularity of the correlation length with critical index ν = 1 and the proportionality of magnetic susceptibility to correlation length in the second order phase transition region of spin-1/2 ferromagnetic models on the Bethe-like lattices are established analytically.
An exact expression for the spin-spin correlation function is derived for the zero-temperature random-field Ising model defined on a Bethe lattice of arbitrary coordination number.The correlation length describing dynamic spinspin correlations and separated from the intrinsic topological length scale of the Bethe lattice is shown to diverge as a power law at the critical point.The critical exponents governing the behaviour of the correlation length are consistent with the mean-field values found for a hypercubic lattice with dimension greater than the upper critical dimension.
We exactly solve the ferromagnetic spin- 1/2 Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green's function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green's functions and the correlation functions are written in terms of a finite set of parameters to be self-consistently determined. A procedure is developed that allows us to exactly fix the unknown parameters in the case of a Bethe lattice with any coordination number z. Nonlocal correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities.
The method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin Sz=±1S^z=\pm1Sz=±1) the expressions for pair and ternary correlation functions within two-particle approximation in { q }\set{q}{q}-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility χ({ q },E)χ(\set{q},E)χ({q},E) is found. In the 1D case and in the absence of external field the expression for χ({ q },E)χ(\set{q},E)χ({q},E) is exact. For the Emery-Blume-Griffiths model (Sz=−2,0,2S^z=-2,0,2Sz=−2,0,2) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility χ({ q })χ(\set{q})χ({q}) of KD2PO4KD_2PO_4KD2​PO4​ ferroelectrics.
We use an effective field model which explicitly incorporates correlations to determine the critical temperatures for Ising ferromagnets on the common lattices. We specifically retain only the first nontrivial odd correlation function, i.e., the triplet correlation function, while neglecting higher order correlations. Within this approximation, assuming the equivalence of triplet correlations involving both central and noncentral sites in the nearest neighbor cluster yields values for the critical temperature below the Bethe approximations for z⩾4. Likewise, assuming the inequivalence of triplet correlations involving the central and noncentral sites yields values of the critical temperature only slightly higher than the Bethe approximation for z⩾4. For lattices with four- and six-fold coordination this equivalence and inequivalence of triplet correlation functions is sufficient to cause the critical temperatures to approach their two- and three-dimensional values. For instance, for quadratic lattices we obtain values for the critical temperature of 2.5284 and 2.9796 as compared to the exact and series results of 2.2692 and 2.7044 for the simple quadratic and diamond lattices, respectively, and the Bethe approximation of 2.8854.
Abstract The expressions for average products of three neighboring spins in the Ising model have been found on lattices with coordination numbers 3 and 4 as functions of temperature and spontaneous magnetization. These expressions are used to compare the exact solution for the Ising model on a square lattice and the solutions found by approximate methods. A method of improving approximate methods is proposed for applying, in particular, to the Bethe approximation and to a change in the critical index of the temperature dependence of the spontaneous magnetization.