The three-dimensional randomly dilute Ising model: Monte Carlo results

TL;DR

This paper presents high-precision Monte Carlo simulations of the 3D randomly dilute Ising model, accurately determining critical exponents, fixed-point couplings, and universal amplitude ratios, with results supporting and refining field-theoretical predictions.

Contribution

It provides the first high-statistics Monte Carlo estimates of critical exponents and fixed-point couplings for the 3D randomly dilute Ising model at a specific density, improving the understanding of its critical behavior.

Findings

Critical exponents: ν=0.683(3), η=0.035(2), β=0.3535(17), α=-0.049(9)

Fixed-point couplings estimated, slightly differing from field theory

Universal amplitude ratio R^+_ξ=0.2885(15)

Abstract

We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices $L^3$ with $L\le 256$. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining $\nu = 0.683(3)$, $\eta = 0.035(2)$, $\beta = 0.3535(17)$, and $\alpha = -0.049(9)$, in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio $R^+_\xi$ that expresses the universality of the free-energy density per correlation volume. We find $R^+_\xi = 0.2885(15)$.