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arXiv:2008.03169 · Physical review. B./Physical review. B

Critical exponent ν of the Ising model in three dimensions with long-range correlated site disorder analyzed with Monte Carlo techniques

Stanislav Kazmin, Wolfhard Janke E

Published Nov 20, 2020• 6 citations
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Abstract

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 $a$ 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 $a$ and the concentration of defects ${p}_{d}$. 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.

DOI
10.1103/physrevb.102.174206

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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{r}^{\ensuremath{-}a}r\ensuremath−a. We derive the critical exponent of the correlation length \ensuremathν\ensuremath{\nu}\ensuremathν and the confluent correction exponent \ensuremathω\ensuremath{\omega}\ensuremathω in dependence of aaa by combining different concentrations of defects 0.05\ensuremath≤pd\ensuremath≤0.40.05\ensuremath{\le}{p}_{d}\ensuremath{\le}0.40.05\ensuremath≤pd​\ensuremath≤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≤a\ensuremath≤3.51.5\ensuremath{\le}a\ensuremath{\le}3.51.5\ensuremath≤a\ensuremath≤3.5 as well as the uncorrelated case a=\ensuremath∞a=\ensuremath{\infty}a=\ensuremath∞ 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: \ensuremathν=2/a\ensuremath{\nu}=2/a\ensuremathν=2/a, and discuss the occurring deviations.
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​.
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.
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.