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arXiv:1908.01880 · Physical review. B./Physical review. B

Three-dimensional universality class of the Ising model with power-law correlated critical disorder

Wenlong Wang, Hannes Meier, Jack Lidmar, Mats Wallin

Published Oct 30, 2019• 9 citations
arXivHTMLPDF

Abstract

We use large-scale Monte Carlo simulations to test the Weinrib-Halperin criterion that predicts new universality classes in the presence of sufficiently slowly decaying power-law correlated quenched disorder. While new universality classes are reasonably well established, the predicted exponents are controversial. We propose a method of growing such correlated disorder using the three-dimensional Ising model as a benchmark system for both generating disorder and studying the resulting phase transition. Critical equilibrium configurations of a disorder-free system are used to define the two-value distributed random bonds with a small power-law exponent given by the pure Ising exponent. Finite-size scaling analysis shows a new universality class with a single phase transition, but the critical exponents ${\ensuremath{\nu}}_{d}=1.13(5),{\ensuremath{\eta}}_{d}=0.48(3)$ differ significantly from theoretical predictions. We find that depending on the details of the disorder generation, disorder-averaged quantities can develop peaks at two temperatures for finite sizes. Finally, a layer model with the two values of bonds spatially separated in halves of the system genuinely has multiple phase transitions, and thermodynamic properties can be flexibly tuned by adjusting the model parameters.

DOI
10.1103/physrevb.100.144204

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Fractals•Feb 1, 2003•0 citations
We use large-scale Monte Carlo simulations to test the Weinrib-Halperin criterion that predicts new universality classes in the presence of sufficiently slowly decaying power-law correlated quenched disorder. While new universality classes are reasonably well established, the predicted exponents are controversial. We propose a method of growing such correlated disorder using the three-dimensional Ising model as a benchmark system for both generating disorder and studying the resulting phase transition. Critical equilibrium configurations of a disorder-free system are used to define the two-value distributed random bonds with a small power-law exponent given by the pure Ising exponent. Finite-size scaling analysis shows a new universality class with a single phase transition, but the critical exponents \ensuremathνd=1.13(5),\ensuremathηd=0.48(3){\ensuremath{\nu}}_{d}=1.13(5),{\ensuremath{\eta}}_{d}=0.48(3)\ensuremathνd​=1.13(5),\ensuremathηd​=0.48(3) differ significantly from theoretical predictions. We find that depending on the details of the disorder generation, disorder-averaged quantities can develop peaks at two temperatures for finite sizes. Finally, a layer model with the two values of bonds spatially separated in halves of the system genuinely has multiple phase transitions, and thermodynamic properties can be flexibly tuned by adjusting the model parameters.
The universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality, is investigated by finite size scaling techniques using the Monte Carlo method. We find that the critical temperature, the critical exponents, and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short-range correlated disordered systems. Our results agree fairly well with theoretical predictions previously made by Weinrib and Halperin for systems with long-range correlated disorder.
We study the critical behavior of the Ising model in three dimensions on a lattice with site disorder by using Monte Carlo simulations. The disorder is either uncorrelated or long-range correlated with correlation function that decays according to a power law ${r}^{\ensuremath{-}a}$. We derive the critical exponent of the correlation length $\ensuremath{\nu}$ and the confluent correction exponent $\ensuremath{\omega}$ in dependence of aaa by combining different concentrations of defects $0.05\ensuremath{\le}{p}_{d}\ensuremath{\le}0.4$ into one global fit ansatz and applying finite-size scaling techniques. We simulate and study a wide range of different correlation exponents $1.5\ensuremath{\le}a\ensuremath{\le}3.5$ as well as the uncorrelated case $a=\ensuremath{\infty}$ and are able to provide a global picture not yet known from previous works. Additionally, we perform a dedicated analysis of our long-range correlated disorder ensembles and provide estimates for the critical temperatures of the system in dependence of the correlation exponent aaa and the concentration of defects pd{p}_{d}pd​. We compare our results to known results from other works and to the conjecture of Weinrib and Halperin: $\ensuremath{\nu}=2/a$, and discuss the occurring deviations.
We present a finite-size scaling analysis of high-statistics Monte Carlo simulations of the three-dimensional randomly site-diluted and bond-diluted Ising model. The critical behaviour of these systems is affected by slowly decaying scaling corrections which make the accurate determination of their universal asymptotic behaviour quite hard, requiring an effective control of the scaling corrections. For this purpose we exploit improved Hamiltonians, for which the leading scaling corrections are suppressed for any thermodynamic quantity, and improved observables, for which the leading scaling corrections are suppressed for any model belonging to the same universality class. The results of the finite-size scaling analysis provide strong numerical evidence that phase transitions in three-dimensional randomly site-diluted and bond-diluted Ising models belong to the same randomly dilute Ising universality class. We obtain accurate estimates of the critical exponents, ν = 0.683(2), η = 0.036(1), α = −0.049(6), γ = 1.341(4), β = 0.354(1), δ = 4.792(6), and of the leading and next-to-leading correction-to-scaling exponents, ω = 0.33(3) and ω 2 = 0.82(8).
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.
The critical behavior of the Ising model in three dimensions on a lattice with site disorder is studied by applying Monte Carlo simulation techniques. Two cases for the site disorder are considered: uncorrelated disorder and long-range correlated disorder with a spatial correlation function that decays according to a power law ${r}^{\ensuremath{-}a}$. The critical exponents $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ as well as updated results for the critical exponent $\ensuremath{\nu}$ and confluent correction exponent $\ensuremath{\omega}$ are provided for a variety of different correlation exponents aaa and disorder concentrations pd{p}_{d}pd​. The estimation is done by using finite-size scaling analyses and a global fit procedure which combines the results obtained for different concentrations of defects. From the estimated critical exponents, the validity of hyperscaling relations is studied and finally the critical temperatures are provided for different aaa and pd{p}_{d}pd​.
In this paper finite size scaling techniques are used to study the universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality. The critical temperature, the critical exponents and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short range correlated disordered systems. This result is in agreement with theoretical predictions previously made by Weinrib and Halperin for systems with long range correlated disorder.