Critical Properties from Monte Carlo Coarse Graining and Renormalization

The distribution function PL(s) of the local order parameter s in finite blocks of size Lᵈ is studied for Ising models for dimensionalities d=2, 3, 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 β and ν for $d=3$ are found as 2βν=1.03±0.01, 1ν=1.60±0.05). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities $n>1$.