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Physical review. B, Condensed matter

Critical properties of random Potts models

Wolfgang Kinzel, Eytan Domany

Published Apr 1, 1981• 131 citations
Publisher

Abstract

The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.

DOI
10.1103/physrevb.23.3421

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The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.
The randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold p∗{p}^{\mathrm{*}}p∗ at which the critical temperature goes to zero depends upon q.
The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for nnn-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.
Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d(1) of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d-d(1)=1 (for which we find non-physical random fixed points, suggesting the existence of non-perturbative fixed distributions) and d-d(1)>1 (for which we do find acceptable perturbative random fixed points), in agreement with previous numerical calculations by Andelman and Aharony [Phys. Rev. B 31, 4305 (1985)]. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.
Critical properties are studied in systems with quenched bond disorder that is correlated along d1{d}_{1}d1​ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+d1{d}_{1}d1​\ensuremath{\nu} is used. For d1{d}_{1}d1​, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For d1{d}_{1}d1​=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for d1{d}_{1}d1​=1 are relevant to random quantum systems.
The randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold pc{p}_{c}pc​. This provides a possible explanation for the apparent specific-heat rounding in certain experiments.