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Journal of Physics C Solid State Physics

Critical specific heat of 2D Ising model in presence of bond defects

John A. Blackman, J. Poulter

Published Jan 10, 1984• 10 citations
Publisher

Abstract

The bond disordered two-dimensional Ising model is studied for a small concentration of randomly substituted defects. The fermion mapping is used and the development is in terms of alloy scattering theory. Three approximations are used. At the lowest level, the ATA reproduces the results of the superlattice model. The CPA is equivalent to the annealed model. An adequate treatment of the quenched model requires the leading contributions from all cross-line diagrams to be included. This is accomplished by parquet theory, and the scattering from three channels is evaluated. Cancellation between two of the channels leaves the third channel as the only contributor. Three conflicting estimates of the specific heat behaviour of the quenched model have appeared in the literature recently. The results of this paper agree with those of Dotsenko and Dotsenko (1982), namely ln(-ln mod T-T c mod ).

DOI
10.1088/0022-3719/17/1/017

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

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Exact critical behavior of a random bond two-dimensional Ising model

R. Shankar

A 2D Ising model in which the bonds K fluctuate randomly about ${\mathrm{K}}_{\mathrm{c}}$, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, ${\mathrm{s}}_{0}$${\mathrm{s}}_{\mathrm{R}}$${\mathrm{〉}}^{2}$${\mathrm{〉}}_{\mathrm{Av}}$, is shown to decay as (lnR${)}^{1/4}$${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/2}$ aat the critical point. This implies that ${\mathrm{s}}_{0}$${\mathrm{s}}_{\mathrm{R}}$${}_{\mathrm{Av}}$ is bounded above by (lnR${)}^{1/8}$${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/4}$ in disagreement with the exp [-(ln lnR${)}^{2}$] 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-${\mathrm{K}}_{\mathrm{c}}$).

Physical Review Letters•Jun 8, 1987•160 citations
The bond disordered two-dimensional Ising model is studied for a small concentration of randomly substituted defects. The fermion mapping is used and the development is in terms of alloy scattering theory. Three approximations are used. At the lowest level, the ATA reproduces the results of the superlattice model. The CPA is equivalent to the annealed model. An adequate treatment of the quenched model requires the leading contributions from all cross-line diagrams to be included. This is accomplished by parquet theory, and the scattering from three channels is evaluated. Cancellation between two of the channels leaves the third channel as the only contributor. Three conflicting estimates of the specific heat behaviour of the quenched model have appeared in the literature recently. The results of this paper agree with those of Dotsenko and Dotsenko (1982), namely ln(-ln mod T-T c mod ).
A study is made of the 2D Ising model in the presence of quenched site disorder at low concentrations. The fermion representation of the Ising model is used and a parquet expansion approach is followed as in earlier work on the bond model by the present authors. A ln(ln(1/ mod T-T c mod )) divergence in the specific heat is found, as in the bond case, demonstrating the expected universality. The authors also discuss the point group symmetry of the fermion representation. Finally the discussion is extended to general defects and it is shown that the specific heat divergence is universal for any type of low-concentration inhomogeneity.
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.
A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -T c -1 dT c /dx mod x=0 =1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -T c -1 dT c /d epsilon 2 mod epsilon =0 =0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C 0 /(1+x gamma 2 C 0 ), where gamma is a constant and C 0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/T c approximately x (1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x -1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.
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​).