Ferromagnetic Potts Model on a Hierarchical Lattice With Random Layered Interactions

Muzy, P. T.; Salinas, S. R.

doi:10.1142/s0217979299000254

Cited by 4 publications

(5 citation statements)

References 9 publications

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“…Using a weak-disorder scheme, we obtain a new (random) fixed point for q larger than a characteristic value q 0 , where disorder becomes relevant. As in a previous publication [10], this fixed point is located in a nonphysical region of the parameter space, suggesting that a nonperturbative fixed point must be present. In Sec.…”

Section: Introductionsupporting

confidence: 79%

“…In the present paper, we use a (perturbative) weak-disorder [9,10] real-space RG scheme to analyze the critical behavior of q-state Potts models with correlated disordered exchange interactions on various hierarchical lattices, whose exact recursion relations are equivalent to those produced by Migdal-Kadanoff approximations for Bravais lattices. Using this weak-disorder scheme, we obtain analytical results by truncating the recursion relations for the moments of the disorder distribution (which are supposed to remain sufficiently small under the RG iterations).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Correlated disordered interactions on Potts models

Muzy¹,

Vieira²,

Salinas³

2002

Phys. Rev. E

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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 d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d − d1 = 1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d − d1 > 1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calculations by Andelman and Aharony. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.

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Section: Introductionsupporting

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Correlated disordered interactions on Potts models

Muzy¹,

Vieira²,

Salinas³

2002

Phys. Rev. E

Self Cite

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“…In particular, in [17], it was shown that hierarchical lattices develop, under frustration, chaotic mapping between different generations, thus proving a first connection between self-similar lattices, frustration and dynamical systems. Similar analysis were also performed on other specific types fractals lattices such as the Hanoi graph (see, for example [25][26][27][28][29][30][31][32][33]). Fractals often appear in the analysis of dynamical systems [34,35].…”

Section: Introductionmentioning

confidence: 84%

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

Alvarez

2024

J. Stat. Mech.

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We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

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“…The papers [19,20,21,22,23,24,25,26], have been important sources of inspiration as far as the present paper is concerned. In [19] the authors derived a discrete dynamical system to analyze the Potts model on the Bethe lattice.…”

Section: Introductionmentioning

confidence: 98%

“…In [21,22] the authors dealt with the same problem from the point of view of the renormalization group discussing the scheme (in)dependence of their results as well. In [23,24,25,26] the authors derived exact recursion relations for the critical behaviors as well as the Fisher and Lee-Yang zeros of the Potts model on various hierarchical lattices: these recursion relations have been analyzed using dynamical system theory.…”

Section: Introductionmentioning

confidence: 99%

Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions

Alvarez¹,

Canfora²,

Parisi³

2010

Preprint

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We present an analytic study of the Potts model partition function on two different types of self-similar lattices of triangular shape with non integer Hausdorff dimension. Both types of lattices analyzed here are interesting examples of non-trivial thermodynamics in less than two dimensions. First, the Sierpinski gasket is considered. It is shown that, by introducing suitable geometric coefficients, it is possible to reduce the computation of the partition function to a dynamical system, whose variables are directly connected to (the arising of) frustration on macroscopic scales, and to determine the possible phases of the system. The same method is then used to analyse the Hanoi graph. Again, dynamical system theory provides a very elegant way to determine the phase diagram of the system. Then, exploiting the analysis of the basins of attractions of the corresponding dynamical systems, we construct various examples of self-similar lattices with more than one critical temperature. These multiple critical temperatures correspond to crossing phases with different degrees of frustration.

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Ferromagnetic Potts Model on a Hierarchical Lattice With Random Layered Interactions

Cited by 4 publications

References 9 publications

Correlated disordered interactions on Potts models

Correlated disordered interactions on Potts models

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions

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