Monte Carlo renormalization-group calculations of critical behavior in the simple-cubic Ising model

Abstract

The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with $8^3$, ${16}^3$, ${32}^3$, and ${64}^3$ sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling $K_1^c=0.221\mathrm{}654\left(6\right)\mathrm{}$. After allowing for (i) interpolation to this best estimate for $K_1^c$, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\nu =0.629\mathrm{}\left(4\right)\mathrm{}$ and $\eta =0.031\mathrm{}\left(5\right)\mathrm{}$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\omega$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.