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arXiv:cond-mat/9711205 · Physical Review Letters

Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model

Pierre Emmanuel Berche, Christophe Chatelain, Bertrand Berche

Published Jan 12, 1998• 28 citations
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Abstract

We investigate the critical behavior of the two-dimensional 8-state Potts model with an aperiodic distribution of the exchange interactions between nearest-neighbor rows. The model is studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm. The transition point is located through duality relations, and the critical behavior is investigated using finite-size scaling techniques at criticality. For strong enough fluctuations of the aperiodic sequence under consideration, a second-order phase transition is found. The exponents $\ensuremath{\beta}/\ensuremath{\nu}$ and $\ensuremath{\gamma}/\ensuremath{\nu}$ are obtained at the new fixed point.

DOI
10.1103/physrevlett.80.297

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We investigate the critical behavior of the two-dimensional 8-state Potts model with an aperiodic distribution of the exchange interactions between nearest-neighbor rows. The model is studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm. The transition point is located through duality relations, and the critical behavior is investigated using finite-size scaling techniques at criticality. For strong enough fluctuations of the aperiodic sequence under consideration, a second-order phase transition is found. The exponents \ensuremathβ/\ensuremathν\ensuremath{\beta}/\ensuremath{\nu}\ensuremathβ/\ensuremathν and \ensuremathγ/\ensuremathν\ensuremath{\gamma}/\ensuremath{\nu}\ensuremathγ/\ensuremathν are obtained at the new fixed point.
Using the ``multihit'' Swendsen-Wang cluster flipping method, we performed extensive Monte Carlo simulations to investigate the critical behavior of the two-dimensional (2D) eight-state random-bond Potts model. We applied finite-size-scaling techniques to extract the critical exponents for two different sets of bond strengths, from which we concluded that the transition is second order with critical exponents for both sets falling into same universality class, that of a 2D Ising model. A variation of the Lee-Kosterlitz method for determining the order of a phase transition was also applied. The double-peaked structure of the specific heat, which was found in some of the bond configurations, was also studied by simulation on periodic arrangements of strong and weak bonds.
The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We can conclusively rule out that the model belongs to the 2D Ising model universality class. The dimensions of the relevant bulk and surface scaling fields are found to be yh=1.849{y}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.849yh​=1.849, yt=0.977{y}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.977yt​=0.977, and yhs=0.54{y}_{{h}_{s}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.54yhs​​=0.54, in contrast to their Ising values 15/815/815/8, 111, and 1/21/21/2, respectively.
We present an extensive Monte Carlo study of the critical relaxation for the two-dimensional square lattice random-bond Potts model using Swendsen-Wang cluster flipping. The integrated autocorrelation time \ensuremath{\tau} is calculated and the dynamic exponent z is estimated by analyzing the size dependence of \ensuremath{\tau}. We find that z\ensuremath{\approx}0 in agreement with estimates for the two-dimensional Ising model. We also present a study of the size dependence of the dynamic behavior of the pure eight-state Potts model which undergoes a first-order transition. The scaling is describable by the product of an exponential times a power law, a behavior which is quite different from that found in the random-bond case.
Abstract The Wang–Landau algorithm of the Monte Carlo method is used to study the magnetic structures of the ground state, phase transitions, and thermodynamic properties of the two-dimensional antiferromagnetic Potts model with number q = 4 of spin states in a hexagonal lattice with the interactions of the nearest (J1) and next-to-nearest (J2) neighbors. The exchange interaction ratios in the range of 0.0 ≤ r = |J2/J1| ≤ 1.0 are studied. A phase diagram of the dependence of the critical temperature on the exchange interaction value of the next-to-nearest neighbors is constructed. The nature of phase transitions is analyzed. It is found that a first-order phase transition is observed in the range of 0.1 ≤ r ≤ 1.0, while there is no phase transition in the system and a strong degeneracy of the ground state is observed at r = 0.0.
We investigate the two-dimensional q=3 and 4 Potts models with a variable interaction range by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges as expressed by the number z of equivalent neighbors. For not-too-large z, the transitions fit well in the universality classes of the short-range Potts models. However, at longer ranges, the transitions become discontinuous. For q=3 we locate a tricritical point separating the continuous and discontinuous transitions near z=80, and a critical fixed point between z=8 and 12. For q=4 the transition becomes discontinuous for z>16. The scaling behavior of the q=4 model with z=16 approximates that of the q=4 merged critical-tricritical fixed point predicted by the renormalization scenario.