Softening of first-order transition in three-dimensions by quenched disorder

Chatelain, Christophe; Berche, Bertrand; Janke, Wolfhard; Berche, Pierre-Emmanuel

doi:10.1103/physreve.64.036120

Cited by 67 publications

(111 citation statements)

References 37 publications

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“…However, there is an important difference. α for the random fixed point is positive and thus ν violates the bound (75) in contrary to recent numerical studies (Ballesteros et al, 2000;Chatelain et al, 2001;Mercaldo et al, 2005;2006;Murtazaev et al, 2007;Yin et al, 2005;2006) and a renormalization-group analysis (Aharony et al, 1998). In the two-dimensional random-bond Potts model, a positive α has also been found as pointed out above.…”

Section: Resultscontrasting

confidence: 40%

Finite-time Scaling and its Applications to Continuous Phase Transitions

Zhong¹

2011

Applications of Monte Carlo Method in Science and Engineering

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Finite-time Scaling and its Applications to Continuous Phase Transitions 18www.intechopen.com forms for the magnetization M, the susceptibility χ, and the specific heat C as M(τ, L)=Lχ(τ, L)=L γ/ν f 2 (τL 1/ν ),using the scaling laws or relationswhere α and γ are critical exponents and the f s including those that will appear later are all scaling functions. In terms of the infinite system correlation length ξ ∞ that diverges at T c asthe argument of f s in Equations (3) is proportional to L/ξ ∞ that governs the finite-size behavior; for small L/ξ ∞ , finite-size scaling appears in which L is a relevant length scale, while large L/ξ ∞ is the thermodynamic limit in which equilibrium behavior shows and L is irrelevant. Note that all the critical exponents assume their infinite-lattices values due to the aforemention assumption (Brézin, 1982;Brézin & Zinn-Justin, 1985). Consequently, measuring the observables for a series of L can then determine the corresponding exponent ratios and finally the critical exponents themselves, the pitch of the critical properties, from the pure power laws emerged exactly at T c or τ = 0 at which f s are assumed to be analytic. In fact, for too small systems sizes and temperatures too far away from T c , corrections to scaling (Wegner, 1972) have to be taken into account. Nevertheless, delicate methods have been developed for extracting critical exponents as well as T c (Amit & Martin-Mayor, 2005;Landau & Binder, 2005). A sequence of Monte Carlo updates may be interpreted as a discrete Markov process (Glauber, 1963;Landau & Binder, 2005;Müler-Krumbhaar & Binder, 1973). Consequently, Monte Carlo simulations can also be applied to study time-dependent dynamic behavior, though usually studied is stochastic relaxational dynamics instead of 'true dynamics' in which the dynamics is determined by the equations of motion derived from a Hamiltonian. Yet, the stochastic dynamics for the kinetic Ising model with local spin dynamics as realized in the single-site Metropolis algorithm (Metropolis et al., 1953), for instance, is believed to fall into the same universality class as that governed by the time-dependent Ginzburg-Landau equation (Hohenberg & Halperin, 1977). Dynamic critical phenomena (Cardy, 1996;Ferrell et al., 1967;Folk & Moser, 2006;Halperin & Hohenberg, 1967;Hohenberg & Halperin, 1977;Ma, 1976) are also companying with a divergent correlation time t eq which diverges with the correlation length ξ ∞ aswith a new dynamical critical exponent z dynamic finite-size scaling (Suzuki, 1977) can be obtained by formally incorporating the time argument t in Equation (1), giving rise to which implies a dynamic finite-size scaling form for the correlation timeTherefore, at the criticality,in the asymptotic region of large time, large size, and small τ. This is again a standard method to estimate z, though when the asymptotic region is reached is not easy to determine (Landau & Binder, 2005;Wansleben & Landau, 1991). However, actual simulations can only be performed inevitably in a limited time ...

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Section: Resultscontrasting

confidence: 40%

Finite-time Scaling and its Applications to Continuous Phase Transitions

Zhong¹

2011

Applications of Monte Carlo Method in Science and Engineering

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“…For d = 2 even infinitesimal disorder induces a continuous transition [3], whereas for d = 3, q > 2 a tricritical point at a finite disorder strength is expected [4]. This softening to a second-order phase transition beyond a tricritical point at some finite disorder strength has recently been verified in Monte Carlo (MC) simulations of the three-dimensional site-diluted 3-state [5] and bond -diluted 4-state [6] Potts model.…”

Section: Introductionmentioning

confidence: 82%

“…It assumes that the expected singularity of the form (15) is the closest to the origin. Then the consecutive ratios of series coefficients behave asymptotically as r n = a n a n−1 = v [6]. For small p, in the first-order region, the series underestimates the critical temperature.…”

Section: Series Analysis: Techniques and Resultsmentioning

confidence: 99%

Star-graph expansions for bond-diluted Potts models

Hellmund

Janke

2003

Phys. Rev. E

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We derive high-temperature series expansions for the free energy and the susceptibility of randombond q-state Potts models on hypercubic lattices using a star-graph expansion technique. This method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions. Moreover, we can keep the disorder strength p as well as the dimension d as symbolic parameters. By applying several series analysis techniques to the new series expansions, one can scan large regions of the (p, d) parameter space for any value of q. For the bond-diluted 4-state Potts model in three dimensions, which exhibits a rather strong first-order phase transition in the undiluted case, we present results for the transition temperature and the effective critical exponent γ as a function of p as obtained from the analysis of susceptibility series up to order 18. A comparison with recent Monte Carlo data (Chatelain et al., Phys. Rev. E64, 036120(2001)) shows signals for the softening to a second-order transition at finite disorder strength.

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“…While this scenario has been rigorously established for the case of two dimensions and an arbitrarily small amount of disorder [52,53,65], the situation for higher-dimensional systems is less clear. For a variety of systems in three dimensions, however, sufficiently strong disorder has been shown numerically [66][67][68] to be able to soften the transition to a continuous one.…”

Section: Harris and Harris-luck Criteriamentioning

confidence: 99%

Two-dimensional quantum gravity - a laboratory for fluctuating graphs and quenched connectivity disorder

Janke¹,

Johnston²,

Weigel³

2006

Condens. Matter Phys.

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This paper gives a brief introduction to using two-dimensional discrete and Euclidean quantum gravity approaches as a laboratory for studying the properties of fluctuating and frozen random graphs in interaction with "matter fields" represented by simple spin or vertex models. Due to the existence of numerous exact analytical results and predictions for comparison with simulational work, this is an interesting and useful enterprise.

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Softening of first-order transition in three-dimensions by quenched disorder

Cited by 67 publications

References 37 publications

Finite-time Scaling and its Applications to Continuous Phase Transitions

Finite-time Scaling and its Applications to Continuous Phase Transitions

Star-graph expansions for bond-diluted Potts models

Two-dimensional quantum gravity - a laboratory for fluctuating graphs and quenched connectivity disorder

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