Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

Abstract

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

High-resolution Monte Carlo study of the order-parameter distribution of the three-dimensional Ising model

“…Although the estimates by Eqs. (6,7) are higher than those from our MC data and Ref. [18], they still agree with each other to within two error bars.…”

“…(4,5) and Eqs. (6,7) are used together. Error bars are estimated by using the propagation of uncertainty with correlation included (covariances between parameters a, b, and c are taken into account).…”

“…where i denotes each of the N spins, and −1 ≤ m ≤ 1. (color online) Scaled probability distribution P (m)L −β/ν as a function of mL β/ν at the critical point Kc = 0.221 654 626 [7]. Curves from the top to the bottom in the inset correspond to lattice sizes L = 16, 32, 96, and 256 respectively.…”

“…1 shows the scaled probability distribution P (m)L −β/ν as a function of mL β/ν at the critical point K c = 0.221 654 626 for finite lattice sizes (L = 16, 32, 96, and 256). Here, β and ν are critical exponents for infinite lattices, and β/ν = 0.518 01(35) [7]. (We used this estimate in our analysis for consistency since this work uses the same data as that of Ref.…”

“…(We used this estimate in our analysis for consistency since this work uses the same data as that of Ref. [7].) The values of the scaled peaks P (m)L −β/ν decreases as the lattice size L increases.…”