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Journal of Physics A Mathematical and General

Critical behaviour of the Baxter model with impurity lattice bonds

Виктор Степанович Доценко, Vl.S. Dotsenko

Published Apr 1, 1984• 22 citations
Publisher

Abstract

From one point of view, the Baxter model (1971) is a model of two coupled 2D Ising lattices. It is known that the specific heat exponent alpha in this model is proportional to the coupling between the lattices g, for small coupling: alpha approximately=4g/ pi <<1. The critical behaviour of the specific heat of the Baxter model with impurity lattice bonds is studied and it is found that the Harris criterion (1974) does not hold in this case. In particular for alpha approximately g<0 the critical specific heat of the pure model C pure ( pi ) approximately - mod tau mod 4(g) pi /( tau =(T-T c )/T c ) changes to a function with stronger cusp singularity C imp ( tau ) approximately -(ln ln 1/ mod tau mod ) -1 , while according to the Harris criterion it should not change in this case. For alpha approximately g>0 the change is from C pure ( tau ) approximately mod tau mod -4g pi / to C imp ( tau ) approximately ln ln 1/ mod tau mod .

DOI
10.1088/0305-4470/17/5/016

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From one point of view, the Baxter model (1971) is a model of two coupled 2D Ising lattices. It is known that the specific heat exponent alpha in this model is proportional to the coupling between the lattices g, for small coupling: alpha approximately=4g/ pi <<1. The critical behaviour of the specific heat of the Baxter model with impurity lattice bonds is studied and it is found that the Harris criterion (1974) does not hold in this case. In particular for alpha approximately g<0 the critical specific heat of the pure model C pure ( pi ) approximately - mod tau mod 4(g) pi /( tau =(T-T c )/T c ) changes to a function with stronger cusp singularity C imp ( tau ) approximately -(ln ln 1/ mod tau mod ) -1 , while according to the Harris criterion it should not change in this case. For alpha approximately g>0 the change is from C pure ( tau ) approximately mod tau mod -4g pi / to C imp ( tau ) approximately ln ln 1/ mod tau mod .
The critical behaviour of the specific heat for the 2D Ising model with impurity bonds is found exactly by the method of Feynman's wanderings over a 2D lattice. In the critical region the model is equivalent to the spinar model with interaction-a zero-component Gross-Neveu model. The specific heat behaves like C approximately ln ln 1/ mod tau mod , as compared with C approximately -ln mod tau mod for the pure model. The slope of the phase transition curve T c ( nu ) near zero concentration of impurity bonds ( nu =0) is also found.
The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.
A 2D Ising model in which the bonds K fluctuate randomly about Kc{\mathrm{K}}_{\mathrm{c}}Kc​, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, s0{\mathrm{s}}_{0}s0​sR{\mathrm{s}}_{\mathrm{R}}sR​〉2{\mathrm{〉}}^{2}〉2〉Av{\mathrm{〉}}_{\mathrm{Av}}〉Av​, is shown to decay as (lnR)1/4{)}^{1/4})1/4${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/2}$ aat the critical point. This implies that s0{\mathrm{s}}_{0}s0​sR{\mathrm{s}}_{\mathrm{R}}sR​Av{}_{\mathrm{Av}}Av​ is bounded above by (lnR)1/8{)}^{1/8})1/8${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/4}$ in disagreement with the exp [-(ln lnR)2{)}^{2})2] decay law found by Dotsenko and Dotsenko by a different method. On the other hand, the present calculation reproduces their specific-heat singularity C\ensuremath{\sim}ln\ensuremath{\Vert}ln\ensuremath{\tau}\ensuremath{\Vert} (\ensuremath{\tau}=K-Kc{\mathrm{K}}_{\mathrm{c}}Kc​).
PACS. 05.50 – Lattice theory and statistics; Ising problems PACS. 75.10 – General theory and models of magnetic ordering We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to 10 6, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents β and γ, caused by impurities. The M and χ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the The problem of inhomogenities influence on phase transitions has a long history. The first model with some kind of impurities to be studied, is, due to its simplicity, the Ising model. Effect of randomness on the critical behaviour of different Ising
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.