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Physical review. B, Condensed matter

Finite-size-scaling analysis of subsystem data in the dilute Ising model

Michael Hennecke

Published Sep 1, 1993• 20 citations
Publisher

Abstract

Monte Carlo simulation results for the magnetization of subsystems of finite lattices are used to determine the critical temperature and a critical exponent of the simple-cubic Ising model with quenched site dilution, at a concentration of p=40%. Particular attention is paid to the effect of the finite size of the systems from which the subsystem results are obtained. This finiteness of the lattices involved is shown to be a source of large deviations of critical temperatures and exponents estimated from subsystem data from their values in the thermodynamic limit. By the use of different lattice sizes, the results ${\mathit{T}}_{\mathit{c}}$(40%)=1.209\ifmmode\pm\else\textpm\fi{}0.002 and \ensuremath{\nu}(40%)=0.78\ifmmode\pm\else\textpm\fi{}0.01 could be extrapolated.

DOI
10.1103/physrevb.48.6271

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Monte Carlo simulation results for the magnetization of subsystems of finite lattices are used to determine the critical temperature and a critical exponent of the simple-cubic Ising model with quenched site dilution, at a concentration of p=40%. Particular attention is paid to the effect of the finite size of the systems from which the subsystem results are obtained. This finiteness of the lattices involved is shown to be a source of large deviations of critical temperatures and exponents estimated from subsystem data from their values in the thermodynamic limit. By the use of different lattice sizes, the results Tc{\mathit{T}}_{\mathit{c}}Tc​(40%)=1.209\ifmmode\pm\else\textpm\fi{}0.002 and \ensuremath{\nu}(40%)=0.78\ifmmode\pm\else\textpm\fi{}0.01 could be extrapolated.
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.
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).
The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.
We study the effects of dilution to the critical properties of site-diluted Ising model in two dimensions using Monte Carlo simulations. Quenched disorder from the dilution is incorporated into the Ising model via random empty sites on the square lattice of Ising spins. Thermodynamic quantities such as the magnetization MMM per spin, energy EEE per spin, magnetic susceptibility χχχ per spin, and specific heat CCC per spin are then calculated after the system has equilibrated. At small dilution concentrations d<0.1d<0.1d<0.1, we find that the value of the critical exponent βββ does not deviate from its pure Ising value. At higher dilution concentrations d>0.1d>0.1d>0.1, however, we find βββ to strongly depend on the value of ddd. We are able to locate a critical temperature TcT_cTc​ and a critical dilution concentration dcd_cdc​ where the phase transition occurs. We find TcT_cTc​ to depend linearly on ddd. In the phase diagrams of MMM, EEE, χχχ, and CCC, we find that the phase transition line eventually disappears at high dilutions. Our results suggest that there is a transition from Strong Universality at low dilution to Weak Universality at high dilution. Lastly, we find a wide and nonsingular increase in the magnetic susceptibility χχχ at the low temperature and high dilution region.