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arXiv:1404.6431 · Condensed Matter Physics

Stability of the Griffiths phase in a 2D Potts model with correlated disorder

Christophe Chatelain

Published Sep 1, 2014• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.5488/cmp.17.33601

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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 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 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.
The Griffiths phase in systems with quenched disorder occurs below the ordering transition of the pure system down to the ordering transition of the actual disordered system. While it does not exhibit long-range order, large fluctuations in the disorder degrees of freedom result in exponentially rare, long-range ordered states and hence the occurrence of broad distributions in response functions. Inside the Griffiths phase of the two-dimensional bond-diluted Ising model the distribution of the magnetic susceptibility is expected to have such a broad, exponential tail. A large-deviation Monte Carlo algorithm is used to sample this distribution and the exponential tail is extracted over a wide range of the support down to very small probabilities of the order of 10^{-300}. We study the behavior of the susceptibility distribution across the full phase diagram, from the paramagnetic state through the Griffiths phase to the ferromagnetically ordered system and down to the zero-temperature point. We extract the rate function of large-deviation theory as well as its finite-size scaling behavior and we reveal interesting differences and similarities between the cases. A connection between the fraction of ferromagnetic bonds in a given disorder sample and the size of the magnetic susceptibility is demonstrated numerically.
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.
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.
Abstract Phase transitions and critical behavior in the 2D dilute Potts model with number q = 4 of spin states on a square lattice are investigated by computer simulation. Systems with linear sizes L × L = N, l = 10–160 with spin concentrations p = 1.00, 0.95, 0.80 are considered. It is shown that the introduction of magnetic impurities plays a stabilizing role in the realization of a second-order phase transition in the Potts model under investigation. Based on the finite-size scaling theory, statistical critical indices of heat capacity α, susceptibility γ, magnetization β, and correlation radius index ν are calculated for the range of concentrations p considered here.