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International Journal of Modern Physics C

THE STUDY OF PHASE TRANSITION BY PERIODIC DISTRIBUTION OF BIMODAL BONDS IN 2D POTTS MODEL

Fatih Yaşar, M. Dilaver

Published Feb 1, 2009• 1 citation
Publisher

Abstract

We have studied the influence of the distribution of bimodal bonds on the phase transition in two-dimensional 8-state Potts model by the recently proposed Wang–Landau (WL) and the Swendsen–Wang (SW) algorithm. All simulations and measurements are done for r = 0.5. Physical quantities such as energy density and specific heat are evaluated at all temperatures. We have also obtained the probability distributions of the energy in order to monitor the transitions. We have observed that some cases of the periodically arranged bond distributions show a single peak, and some cases show double or triple peaks in the specific heat. Besides, it seems that the appearing of these peaks in the specific heat relates to a blocking procedure for periodicity. When the number of interaction pairs between the bimodal bonds is increased on the lattice with the blocking procedure, one can observe a single peak, otherwise, one can observe a double or triple peaks in the specific heat. From the point of view of simulation methods, the WL algorithm also works efficiently in the simulation of the system for a periodically arranged bond distribution as well as the SW algorithm.

DOI
10.1142/s012918310901356x

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We have studied the influence of the distribution of bimodal bonds on the phase transition in two-dimensional 8-state Potts model by the recently proposed Wang–Landau (WL) and the Swendsen–Wang (SW) algorithm. All simulations and measurements are done for r = 0.5. Physical quantities such as energy density and specific heat are evaluated at all temperatures. We have also obtained the probability distributions of the energy in order to monitor the transitions. We have observed that some cases of the periodically arranged bond distributions show a single peak, and some cases show double or triple peaks in the specific heat. Besides, it seems that the appearing of these peaks in the specific heat relates to a blocking procedure for periodicity. When the number of interaction pairs between the bimodal bonds is increased on the lattice with the blocking procedure, one can observe a single peak, otherwise, one can observe a double or triple peaks in the specific heat. From the point of view of simulation methods, the WL algorithm also works efficiently in the simulation of the system for a periodically arranged bond distribution as well as the SW algorithm.
Monte Carlo simulations using the recently proposed Wang–Landau algorithm are performed to the q = 8 state Potts model in two dimension with various degrees of randomness. We systematically studied the effect of quenched bond randomness to system which has first-order phase transition. All simulations and measurements were done from pure case r = 1 to r = 0.4. Physical quantities such as energy density and ground-state entropy were evaluated at all temperatures. We have also obtained probability distributions of energy to monitor softening of transitions. It appears quite feasible to simulate spin systems with quenched bond randomness by Wang–Landau algorithm.
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.
Randomness is an important subject in the study of phase transition as defect and impurity may present in any real material. The pre-existing ordered phase of a pure system can be affected or even ruined by the presence of randomness. Here we study ferromagnetic Ising model on a square lattice with a presence of randomness in the form of bond dilution. The pure system of this model is known to experience second order phase transition, separating between the high temperature paramagnetic and low-temperature ferromagnetic phase. We used Wang-Landau algorithm of Monte Carlo method to obtain the density of states from which we extract the ensemble average of energy and the specific heat. We observed the signature of phase transition indicated by the diverging peak of the specific heat as system sizes increase. These peaks shift to the lower temperature side as the dilution increases. The lower temperature ordered phase preserves up to certain concentration of dilution and is totally ruined when the bonds no longer percolates.
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.
The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.