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arXiv:cond-mat/0201575 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlated disordered interactions on Potts models

Paulo de Tarso Muzy, André P. Vieira, Silvio R. A. Salinas

Published Apr 2, 2002• 5 citations
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Abstract

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.

DOI
10.1103/physreve.65.046120

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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.
We consider the critical behavior of the random qqq-state Potts model in the large-qqq limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin-glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy P(I)P(I)P(I) which is shown to satisfy different integral equations at the fixed points. By numerical integration we have obtained the corresponding stable laws of nonlinear combination of random numbers and obtained numerically exact values for the critical exponents.
We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains (L< 16)(L<~16)(L< 16) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.
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 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 $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.
Phase transitions in a two-dimensional q-state Potts model with long-range disorder in which the correlation decays as a power law approximately r -a for separation r are studied analytically within a Migdal-Kadanoff renormalization group. Correlation terms are introduced into the parameter space of the renormalization group and the recursion relations are derived. We expand q near q 0 , where the specific-heat exponent of the pure system alpha p vanishes, and a near d=2. For small alpha p and delta =2-a<<1, we find three fixed points: 'pure','short-range disorder' and 'long-range disorder'. In our calculation the disorder is relevant if av-2<0 for a<d, where the v are the correlation-length exponents of the fixed points. The correlation-length exponent for the 'long-range disorder' fixed point is v long =2/a. These are consistent with field-theoretic renormalization group results.