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Journal of Physics C Solid State Physics

Effect of random defects on the critical behaviour of Ising models

A. Brooks Harris

Published May 7, 1974• 1,996 citations

Abstract

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.

DOI: 10.1088/0022-3719/7/9/009

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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 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.

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

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 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 ).