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arXiv:cond-mat/0211543 · Physical Review Letters

Phase Transition in the 2D Random Potts Model in the Large- q Limit

J C Anglès d'Auriac, Ferenc Iglói

Published May 13, 2003• 32 citations
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Abstract

Phase transition in the two-dimensional q-state Potts model with random ferromagnetic couplings is studied in the large-q limit by a combinatorial optimization algorithm and by approximate mappings. We conjecture that the critical behavior of the model is controlled by the isotropic version of the infinite randomness fixed point of the random transverse-field Ising spin chain and the critical exponents are exactly given by beta=(3-sqrt[5])/4, beta(s)=1/2, and nu=1. The specific heat has a logarithmic singularity, but at the transition point there are very strong sample-to-sample fluctuations. Discretized randomness results in discontinuities in the internal energy.

DOI
10.1103/physrevlett.90.190601

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The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.
We study phase transitions of the ferromagnetic qqq-state Potts chain with random nearest-neighbor couplings having a variance ${\mathrm{\ensuremath{\Delta}}}^{2}$ and with homogeneous long-range interactions, which decay with distance as a power ${r}^{\ensuremath{-}(1+\ensuremath{\sigma})}, \ensuremath{\sigma}>0$. In the large-qqq limit the free-energy of random samples of length $L\ensuremath{\le}2048$ is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for $\ensuremath{\sigma}<{\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})\ensuremath{\le}0.5$, while the correlation length becomes divergent at the transition point for ${\ensuremath{\sigma}}_{c}(\mathrm{\ensuremath{\Delta}})<\ensuremath{\sigma}<1$. In the latter regime the average magnetization is continuous for small enough $\mathrm{\ensuremath{\Delta}}$, but for larger $\mathrm{\ensuremath{\Delta}}$---according to the numerical results---it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.
Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in 2+1D. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of q, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be ν=1, saturating a modified version of the Harris-Luck criterion.
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.
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We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal properties of which are related to the critical singularities of the random-Potts model. The optimization problem of finding the dominant graph, is studied on the square lattice by simulated annealing and by a combinatorial algorithm. Critical exponents of the magnetization and the correlation length are estimated and conformal predictions are compared with numerical results.