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arXiv:1303.1991 · Europhysics Letters (EPL)

Hyperscaling violation in the 2D 8-state Potts model with long-range correlated disorder

Christophe Chatelain

Published Jun 1, 2013• 8 citations
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Abstract

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.

DOI
10.1209/0295-5075/102/66007

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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 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.
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.
We consider the critical behavior of two-dimensional Potts models in the presence of a bond disorder in which the correlation decays as a power law. In some recent work the thermal sector of this theory was investigated by a renormalization group computation based on perturbed conformal field theory. Here we apply the same approach to study instead the magnetic sector. In particular, we compute the leading corrections to the Potts spin scaling dimension. Our results include as a special case the long-range disorder Ising model. We compare our prediction to Monte Carlo simulations. Finally, by studying the magnetization scaling function, we show a clear numerical evidence of a cross-over between the long-range and the short-range class of universality.
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.
The critical behavior of the Ising model in three dimensions on a lattice with site disorder is studied by applying Monte Carlo simulation techniques. Two cases for the site disorder are considered: uncorrelated disorder and long-range correlated disorder with a spatial correlation function that decays according to a power law ${r}^{\ensuremath{-}a}$. The critical exponents $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ as well as updated results for the critical exponent $\ensuremath{\nu}$ and confluent correction exponent $\ensuremath{\omega}$ are provided for a variety of different correlation exponents aaa and disorder concentrations pd{p}_{d}pd​. The estimation is done by using finite-size scaling analyses and a global fit procedure which combines the results obtained for different concentrations of defects. From the estimated critical exponents, the validity of hyperscaling relations is studied and finally the critical temperatures are provided for different aaa and pd{p}_{d}pd​.
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.