Critical Properties from Monte Carlo Coarse Graining and Renormalization

The distribution function $P_L\mathrm{}\left(s\right)\mathrm{}$ of the local order parameter $s$ in finite blocks of size $L^d$ is studied for Ising models for dimensionalities $d=2, 3, \mathrm{and} 4$ by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small $L$ (e.g., the standard exponents $\beta$ and $\nu$ for $d=3\mathrm{}$ are found as $\frac{2\beta }{\nu }=1.03\mathrm{}\pm 0.01$, $\frac{1}{\nu }=1.60\mathrm{}\pm 0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities $n>\mathrm{}1\mathrm{}$.