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arXiv:1308.0734 · Physical Review E

Griffiths phase and critical behavior of the two-dimensional Potts models with long-range correlated disorder

Christophe Chatelain

Published Mar 10, 2014• 21 citations
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Abstract

The q-state Potts model with long-range correlated disorder is studied by means of large-scale Monte Carlo simulations for q=2, 4, 8, and 16. Evidence is given of the existence of a Griffiths phase, where the thermodynamic quantities display an algebraic finite-size scaling, in a finite range of temperatures. The critical exponents are shown to depend on both the temperature and the exponent of the algebraic decay of disorder correlations, but not on the number of states of the Potts model. The mechanism leading to the violation of hyperscaling relations is observed in the entire Griffiths phase.

DOI
10.1103/physreve.89.032105

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Physical Review E•Jan 22, 2009•8 citations
The q-state Potts model with long-range correlated disorder is studied by means of large-scale Monte Carlo simulations for q=2, 4, 8, and 16. Evidence is given of the existence of a Griffiths phase, where the thermodynamic quantities display an algebraic finite-size scaling, in a finite range of temperatures. The critical exponents are shown to depend on both the temperature and the exponent of the algebraic decay of disorder correlations, but not on the number of states of the Potts model. The mechanism leading to the violation of hyperscaling relations is observed in the entire Griffiths phase.
The first-order phase transition of the two-dimensional eight-state Potts model is shown to be rounded when long-range correlated disorder is coupled to energy density. Critical exponents are estimated by means of large-scale Monte Carlo simulations. In contrast to uncorrelated disorder, a violation of the hyperscaling relation γ/ν=d−2xσ\gamma/\nu=d-2x_\sigmaγ/ν=d−2xσ​ is observed. A simple and general mechanism is presented as to how strong disorder fluctuations cause this violation. In the thermal sector too, evidences are given for such violation in the two hyperscaling relations α/ν=d−2xϵ\alpha/\nu=d-2x_\epsilonα/ν=d−2xϵ​ and 1/ν=d−xϵ1/\nu=d-x_\epsilon1/ν=d−xϵ​. The hyperscaling violation exponent θ\thetaθ is {\sl a-priori} different for these three relations. The scaling dimension of energy is conjectured to be xϵ=a/2x_\epsilon=a/2xϵ​=a/2, where aaa is the exponent of the algebraic decay of disorder correlations.
A Griffiths phase has recently been observed by Monte Carlo simulations in the 2D q-state Potts model with strongly correlated quenched random couplings.In particular, the magnetic susceptibility was shown to diverge algebraically with the lattice size in a broad range of temperatures.However, only relatively small lattice sizes could be considered, so one can wonder whether this Griffiths phase will not shrink and collapse into a single point, the critical point, as the lattice size is increased to much larger values.In this paper, the 2D eight-state Potts model is numerically studied for four different disorder correlations.It is shown that the Griffiths phase cannot be explained as a simple spreading of local transition temperatures caused by disorder fluctuations.As a consequence, the vanishing of the latter in the thermodynamic limit does not necessarily imply the collapse of the Griffiths phase into a single point.By contrast, the width of the Griffiths phase is controlled by the disorder strength.However, for disorder correlations decaying slower than 1/r , no cross-over to a more usual critical behavior could be observed as this strength is tuned to weaker values.
The first-order phase transition of the two-dimensional eight-state Potts model is shown to be rounded when long-range correlated disorder is coupled to energy density. Critical exponents are estimated by means of large-scale Monte Carlo simulations. In contrast to uncorrelated disorder, a violation of the hyperscaling relation γ / ν = d − 2 x σ is observed. Even though the system is not frustrated, disorder fluctuations are strong enough to cause this violation in the very same way as in the 3D random-field Ising model. In the thermal sector, too, evidence is given for such violation in the two hyperscaling relations α / ν = d − 2 x ε and 1/ ν = d − x ε . In contrast to the random field Ising model, at least two hyperscaling violation exponents are needed. The scaling dimension of energy is conjectured to be x ε = a /2, where a is the exponent of the algebraic decay of disorder correlations.
The random quantum q-state clock and Potts models are studied in two and three dimensions. The existence of Griffiths phases is tested in the two-dimensional case with q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu three-dimensional classical models with Monte Carlo simulations. For the random Potts model, numerical evidence of the existence of Griffiths phases is given and the finite-size effects are analyzed. For the clock model, the data also suggest the existence of a Griffiths phase but with much larger finite-size effects. The critical point of the random quantum clock model is then studied with the Strong-Disorder Renormalization Group. Evidence is given that, at strong enough disorder, this critical behavior is governed by the same infinite-disorder fixed point as the Potts model, for all the number of states q considered. At weak disorder, our renormalization group method becomes unstable and does not allow us to make conclusions.
We study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.
We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.