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Advances In Physics

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

Виктор Степанович Доценко, Victor S Dotsenko

Published Jan 1, 1983• 285 citations
Publisher

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 − 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.

DOI
10.1080/00018738300101541

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The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.

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Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

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The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.

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Critical behaviour of the Baxter model with impurity lattice bonds

Виктор Степанович Доценко, Vl.S. Dotsenko

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 .

Journal of Physics A Mathematical and General•Apr 1, 1984•22 citations
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.
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.
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 relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
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 .