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arXiv:cond-mat/0511203 · Physical Review E

Critical and tricritical singularities of the three-dimensional random-bond Potts model for large q

Maria Teresa Mercaldo, J C Anglès d'Auriac, Ferenc Iglói

Published Feb 23, 2006• 21 citations
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Abstract

We study the effect of varying strength $\ensuremath{\delta}$ of bond randomness on the phase transition of the three-dimensional Potts model for large $q$. The cooperative behavior of the system is determined by large correlated domains in which the spins point in the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder $\ensuremath{\delta}>{\ensuremath{\delta}}_{t}$ this percolating cluster coexists with a percolating cluster of noncorrelated spins. Such a coexistence is only possible in more than two dimensions. We argue and check numerically that ${\ensuremath{\delta}}_{t}$ is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.10(2)$ and ${\ensuremath{\nu}}_{t}=0.67(4)$. We claim these exponents are $q$ independent for sufficiently large $q$. In the second-order transition regime the critical exponents ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.60(2)$ and ${\ensuremath{\nu}}_{t}=0.73(1)$ are independent of the strength of disorder.

DOI
10.1103/physreve.73.026126

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We study the effect of varying strength \ensuremathδ\ensuremath{\delta}\ensuremathδ of bond randomness on the phase transition of the three-dimensional Potts model for large qqq. The cooperative behavior of the system is determined by large correlated domains in which the spins point in the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder \ensuremathδ>\ensuremathδt\ensuremath{\delta}>{\ensuremath{\delta}}_{t}\ensuremathδ>\ensuremathδt​ this percolating cluster coexists with a percolating cluster of noncorrelated spins. Such a coexistence is only possible in more than two dimensions. We argue and check numerically that \ensuremathδt{\ensuremath{\delta}}_{t}\ensuremathδt​ is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as \ensuremathβt∕\ensuremathνt=0.10(2){\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.10(2)\ensuremathβt​∕\ensuremathνt​=0.10(2) and \ensuremathνt=0.67(4){\ensuremath{\nu}}_{t}=0.67(4)\ensuremathνt​=0.67(4). We claim these exponents are qqq independent for sufficiently large qqq. In the second-order transition regime the critical exponents \ensuremathβt∕\ensuremathνt=0.60(2){\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.60(2)\ensuremathβt​∕\ensuremathνt​=0.60(2) and \ensuremathνt=0.73(1){\ensuremath{\nu}}_{t}=0.73(1)\ensuremathνt​=0.73(1) are independent of the strength of disorder.
Phase transitions induced by varying the strength of disorder in the large- q state Potts model in 3d are studied by analytical and numerical methods. By switching on the disorder, the transition stays of first order, but different thermodynamical quantities display essential singularities. Only for strong enough disorder will the transition be soften into a second-order one, in which case the ordered phase becomes non-homogeneous at large scales, while the non-correlated sites percolate the sample. In the critical regime the critical exponents are found universal: β/ν = 0.60(2) and ν = 0.73(1).
The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the qqq-state Potts model. For large qqq a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for d>2d>2d>2 such systems exhibit a tricritical point with an exponent $\ensuremath{\nu}$ related to those of the random-field model by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\nu}}_{\mathrm{RF}}/(2\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{RF}}\ensuremath{-}{\ensuremath{\beta}}_{\mathrm{RF}})$. In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio $\ensuremath{\beta}/\ensuremath{\nu}$ which varies continuously with qqq, and a weakly varying exponent $\ensuremath{\nu}\ensuremath{\approx}1$. We find strong evidence for the multiscaling of the correlation functions.
We study by extensive Monte Carlo simulations the effect of random bond dilution on the phase transition of the three-dimensional four-state Potts model that is known to exhibit a strong first-order transition in the pure case. The phase diagram in the dilution-temperature plane is determined from the peaks of the susceptibility for sufficiently large system sizes. In the strongly disordered regime, numerical evidence for softening to a second-order transition induced by randomness is given. Here a large-scale finite-size scaling analysis, made difficult due to strong crossover effects presumably caused by the percolation fixed point, is performed.
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 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.