Critical behaviour of the phase transition in the 2D Ising Model with impurities

Abstract

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 − Τ_c/Τ_c to C ˜ ln ln 1/|τ| for τ≪τ_i ˜ 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 σ_Ο> ˜ 1/R^1/4 to <σ _R σ_Ο> ˜ 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.