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Physical review. E

Numerical evidence of a universal critical behavior of two-dimensional and three-dimensional random quantum clock and Potts models

Valentin Anfray, Christophe Chatelain

Published Jul 24, 2023• 3 citations
Publisher

Abstract

The random quantum q-state clock and Potts models are studied in two and three dimensions. The existence of Griffiths phases is tested in the two-dimensional case with q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu three-dimensional classical models with Monte Carlo simulations. For the random Potts model, numerical evidence of the existence of Griffiths phases is given and the finite-size effects are analyzed. For the clock model, the data also suggest the existence of a Griffiths phase but with much larger finite-size effects. The critical point of the random quantum clock model is then studied with the Strong-Disorder Renormalization Group. Evidence is given that, at strong enough disorder, this critical behavior is governed by the same infinite-disorder fixed point as the Potts model, for all the number of states q considered. At weak disorder, our renormalization group method becomes unstable and does not allow us to make conclusions.

DOI
10.1103/physreve.108.014124

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The random quantum q-state clock and Potts models are studied in two and three dimensions. The existence of Griffiths phases is tested in the two-dimensional case with q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu three-dimensional classical models with Monte Carlo simulations. For the random Potts model, numerical evidence of the existence of Griffiths phases is given and the finite-size effects are analyzed. For the clock model, the data also suggest the existence of a Griffiths phase but with much larger finite-size effects. The critical point of the random quantum clock model is then studied with the Strong-Disorder Renormalization Group. Evidence is given that, at strong enough disorder, this critical behavior is governed by the same infinite-disorder fixed point as the Potts model, for all the number of states q considered. At weak disorder, our renormalization group method becomes unstable and does not allow us to make conclusions.
The random quantum qqq-state clock and Potts models are studied in 2 and 3 dimensions. The existence of Griffiths phases is tested in the 2D case with q=6q=6q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu 3D classical modelswith Monte Carlo simulations. No Griffiths phase is found for the clock model. In contrast, numerical evidences of the existence of Griffiths phases in the random Potts model are given and the Finite Size effects are analyzed. The critical point of the random quantum clock model is then studied by Strong-Disorder Renormalization Group. Despite a chaotic behavior of the Renormalization-Group flow at weak disorder, evidences are given that this critical behavior is governed by the same Infinite-Disorder Fixed Point as the Potts model, independently from the number of states qqq.
The random qqq-state quantum Potts model is studied on hypercubic lattices in dimensions 2 and 3 using the numerical implementation of the strong disorder renormalization group introduced by Kovacs and Igl\'oi [Phys. Rev. B 82, 054437 (2010)]. Critical exponents $\ensuremath{\nu},$ df,{d}_{f},df​, and $\ensuremath{\psi}$ at the infinite disorder fixed point are estimated by finite-size scaling for several numbers of states qqq between 2 and 50. When scaling corrections are not taken into account, the estimates of both df{d}_{f}df​ and $\ensuremath{\psi}$ systematically increase with qqq. It is shown, however, that qqq-dependent scaling corrections are present and that the exponents are compatible within error bars, or close to each other, when these corrections are taken into account. This provides evidence of the existence of a superuniversality of all two- and three-dimensional random Potts models.
The q-state Potts model with long-range correlated disorder is studied by means of large-scale Monte Carlo simulations for q=2, 4, 8, and 16. Evidence is given of the existence of a Griffiths phase, where the thermodynamic quantities display an algebraic finite-size scaling, in a finite range of temperatures. The critical exponents are shown to depend on both the temperature and the exponent of the algebraic decay of disorder correlations, but not on the number of states of the Potts model. The mechanism leading to the violation of hyperscaling relations is observed in the entire Griffiths phase.
A Griffiths phase has recently been observed by Monte Carlo simulations in the 2D q-state Potts model with strongly correlated quenched random couplings.In particular, the magnetic susceptibility was shown to diverge algebraically with the lattice size in a broad range of temperatures.However, only relatively small lattice sizes could be considered, so one can wonder whether this Griffiths phase will not shrink and collapse into a single point, the critical point, as the lattice size is increased to much larger values.In this paper, the 2D eight-state Potts model is numerically studied for four different disorder correlations.It is shown that the Griffiths phase cannot be explained as a simple spreading of local transition temperatures caused by disorder fluctuations.As a consequence, the vanishing of the latter in the thermodynamic limit does not necessarily imply the collapse of the Griffiths phase into a single point.By contrast, the width of the Griffiths phase is controlled by the disorder strength.However, for disorder correlations decaying slower than 1/r , no cross-over to a more usual critical behavior could be observed as this strength is tuned to weaker values.
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 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.