Journal ArticleDOI
Effect of random defects on the critical behaviour of Ising models
TLDR
In this article, a cumulant expansion is used to calculate the transition temperature of simple-square Ising models with random-bond defects, and the results are -Tc-1 dTc/dx mod x=0.329 compared with the mean-field value of one.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 -Tc-1 dTc/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 -Tc-1 dTc/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 C0/(1+x gamma 2C0), where gamma is a constant and C0 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/Tc 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.read more
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The Potts model
TL;DR: In this paper, a tutorial review on the Potts model is presented aimed at bringing out the essential and important properties of the standard Potts models, focusing on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective.
Book
Scaling and Renormalization in Statistical Physics
TL;DR: In this article, the authors provide a thoroughly modern graduate-level introduction to the theory of critical behavior, including phase diagrams, fixed points, cross-over behavior, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry.
Journal ArticleDOI
Localization: theory and experiment
B Kramer,A MacKinnon +1 more
TL;DR: The transport properties of disordered solids have been the subject of much work since at least the 1950s, but with a new burst of activity during the 1980s which has survived up to the present day as mentioned in this paper.
Journal ArticleDOI
Fermi-liquid instabilities at magnetic quantum phase transitions
TL;DR: In this article, the authors discuss the instabilities of the Fermi-liquid state of conduction electrons in metals with particular emphasis on magnetic quantum critical points, with the aim of assessing the validity of presently available theory.
Journal ArticleDOI
Percolation, statistical topography, and transport in random media
TL;DR: A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media as discussed by the authors, where a geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.
References
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Journal ArticleDOI
Crystal statistics. I. A two-dimensional model with an order-disorder transition
TL;DR: In this article, the eigenwert problem involved in the corresponding computation for a long strip crystal of finite width, joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum.
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Scaling laws for ising models near T c
TL;DR: In this paper, a model for describing the behavior of Ising models very near to the homogeneity of the free energy is introduced. The model is based upon dividing the Ising model into cells which are microscopically large but much smaller than the coherence length and then using the total magnetization within each cell as a collective variable.
Journal ArticleDOI
Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet
TL;DR: In this paper, it was shown that in a class of randomly diluted Ising ferromagnets the magnetization fails to be an analytic function of the field at a range of temperatures above that at which spontaneous magnetization first appears.
Journal ArticleDOI
Theory of Critical-Point Scattering and Correlations. I. The Ising Model
TL;DR: The theory of the correlations and critical scattering of two-dimensional nearest-neighbor Ising models is discussed in this paper, where a distinction is drawn between the true inverse range of exponential decay of correlations, and the effective range determined from the low-angle scattering intensity.
Journal ArticleDOI
Critical Percolation Probabilities by Series Methods
M. F. Sykes,J. W. Essam +1 more
TL;DR: In this paper, series estimates of the critical percolation probabilities for the "bond problem" and the "site problem" are presented for two-and three-dimensional lattices.