Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.01176 · Physical review. E

Dynamical critical properties of the random-bond three-state Potts model

Anonymous, Ulvi Kanbur, Mohammad Sadra Najafi, Muktish Acharyya, Erol Vatansever

Published Aug 13, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/7hy2-c47n

Related papers

Critical dynamics of cluster algorithms in the random-bond Ising model

Ulvi Kanbur, Zeynep Demir Vatansever

In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.

Physical review. E•Feb 28, 2024•2 citations

Universality in the critical dynamics of the three-dimensional large-q Potts model with bond randomness

Wanjie Xiong, Chudong Xu

The critical behavior of the three-dimensional random-bond Potts model for large state numbers with q=16 and q=32 is studied by the finite-time dynamics combining with Monte Carlo renormalization group method. The correlation length and dynamic exponents as well as the critical temperatures are calculated through reducing the influence of finite-size effects. According to the variations of critical exponents with disorder amplitude, we identify the asymptotic critical exponents with ν=0.424(14), z=3.03(12) and find that they are q-independent. The results further confirm that the intrinsic ν, instead of finite-size one, can escape the bound ν>2/d suggested previously.

Phase Transitions•Mar 11, 2013•1 citation

Critical dynamics of the two-dimensional random-bond Potts model with nonequilibrium Monte Carlo simulations

Shuangli Fan, Fan Zhong

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.

Physical Review E•Jan 22, 2009•8 citations

Single-cluster dynamics for the random-cluster model

Youjin Deng, Xiaofeng Qian, Henk W. J. Blöte

We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.

arXiv:0907.1711•Physical Review E•Sep 30, 2009•5 citations

Weak Universality in the Disordered Two-Dimensional Antiferromagnetic Potts Model on a Triangular Lattice

Albert Babaevich Babaev, Akai Kurbanovich Murtazaev

The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality $$\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = d$$ . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.

Journal of Experimental and Theoretical Physics Letters•May 1, 2018•5 citations

Investigation of the critical behavior of the three-dimensional weakly diluted Potts model

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, G. Ya. Aznaurova

The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v .

Bulletin of the Russian Academy of Sciences Physics•May 1, 2007•1 citation
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.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.
The critical behavior of the three-dimensional random-bond Potts model for large state numbers with q=16 and q=32 is studied by the finite-time dynamics combining with Monte Carlo renormalization group method. The correlation length and dynamic exponents as well as the critical temperatures are calculated through reducing the influence of finite-size effects. According to the variations of critical exponents with disorder amplitude, we identify the asymptotic critical exponents with ν=0.424(14), z=3.03(12) and find that they are q-independent. The results further confirm that the intrinsic ν, instead of finite-size one, can escape the bound ν>2/d suggested previously.
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.
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality 2βν+γν=d\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = dν2β​+νγ​=d . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.
The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v .