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Journal of Physics C Solid State Physics

Two-point correlation function of the 2D Ising model with impurity lattice bonds

Виктор Степанович Доценко, Vl.S. Dotsenko

Published Jun 20, 1982• 20 citations
Publisher

Abstract

The two-point correlation function of the 2D Ising model with small concentration of impurity bonds ( nu imp <<1) is found to change its critical behaviour from ( sigma r sigma 0 ) approximately 1/R 1/4 for r<<R nu exp(c 1 / nu imp ) to ( sigma R delta 0 ) approximately exp<or=-(c 2 / nu imp )(ln ln R) 2 ) for R>>R, which implies the change of the critical index from 1 / 4 in the pure Ising model to 0 in the model with impurities.

DOI
10.1088/0022-3719/15/17/003

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The two-point correlation function of the 2D Ising model with small concentration of impurity bonds ( nu imp <<1) is found to change its critical behaviour from ( sigma r sigma 0 ) approximately 1/R 1/4 for r<<R nu exp(c 1 / nu imp ) to ( sigma R delta 0 ) approximately exp<or=-(c 2 / nu imp )(ln ln R) 2 ) for R>>R, which implies the change of the critical index from 1 / 4 in the pure Ising model to 0 in the model with impurities.
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.
High accuracy Monte Carlo simulation results for 1024*1024 Ising system with ferromagnetic impurity bonds are presented. Spin-spin correlation function at a critical point is found to be numerically very close to that of a pure system. This is not trivial since a critical temperature for the system with impurities is almost two times lower than pure Ising TcT_cTc​. Small deviations from the pure behaviour contradict to some of the competing theories.
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.
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 .