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Physical review. B./Physical review. B

Critical exponents of the Ising model in three dimensions with long-range power-law correlated site disorder: A Monte Carlo study

Stanislav Kazmin, Wolfhard Janke E

Published Jun 27, 2022• 15 citations
Publisher

Abstract

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 $a$ and disorder concentrations ${p}_{d}$. 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 $a$ and ${p}_{d}$.

DOI
10.1103/physrevb.105.214111

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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{r}^{\ensuremath{-}a}r\ensuremath−a. The critical exponents \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathγ\ensuremath{\gamma}\ensuremathγ as well as updated results for the critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν and confluent correction exponent \ensuremathω\ensuremath{\omega}\ensuremathω 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​.
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 used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.
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.
Monte Carlo simulations of the short-time dynamic behaviour are reported for three-dimensional Ising model and XY-model with long-range spatially correlated disorder at criticality, in the case corresponding to linear defects. The static and dynamic critical exponents are computed with the use of the corrections to scaling. The obtained values of the exponents are in a good agreement with results of the field-theoretic description of the critical behaviour of this model in the two-loop approximation and with our results of Monte Carlo simulations of three-dimensional Ising model in equilibrium state.