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arXiv:cond-mat/9912463 · International Journal of Modern Physics B

CORRELATION FUNCTIONS OF MULTISITE INTERACTION SPIN-S MODELS ON THE BETHE-LIKE LATTICES

Ruben G. Ghulghazaryan

Published Mar 10, 2000• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217979200000534

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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.
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 $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and 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.
Two-layer Bethe lattice whose lattice sites are occupied with spin-3/2 atoms is solved exactly by using the recursion relations in a pairwise approach for given coordination numbers q = 3, 4 and 6 with equal external magnetic fields acting on the layers.The ferromagnetic (FM) and antiferromagnetic (AFM) interactions for the spins of the upper and lower layers, respectively, and either FM or AFM type interactions between the adjacent spins of the layers are assumed.The phase diagrams of the model are studied on different planes for given system parameters by obtaining the ground state (GS) phase diagrams and the thermal variations of the order parameters and the response functions, i. e. the susceptibility and the specific heat, in detail.It was found that the model presents both second-and first-order phase transitions.The reentrant behavior is seen when the model presents two N éel temperatures for higher q values.The existence of the tricritical point and critical end points is also confirmed.
We consider a general spin-1/2 Ising model with multisite interaction on the Husimi lattice with the coordination number q and derive an analytical expression of correlation functions for stable fixed points of the corresponding recurrence relation. We show that for q=2 our model transforms to the two-state vertex model on the Bethe lattice with q=3 and for the case q=3, with only nearest neighbor interactions, we transform our model to the corresponding model on the Bethe lattice with q=3, using the Yang–Baxter equations.
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 compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where ttt is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-ttt behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
Expressions for general correlation functions in Ising ferromagnets are derived in terms of the Lee-Yang theorem. By the use of these expressions, the asymptotic forms of correlation functions in the presence of magnetic field are analyzed, assuming those in the absence of the external field. In particular, the critical indices of spin-spin, energy-density-energy-density, energy-density-spin correlation functions are related with the critical indices of thermal susceptiblity χT0, specific heat, susceptibility χH0, and magnetization. These results agree in form with those predicted by the scaling law approach.