Critical Properties from Monte Carlo Coarse Graining and Renormalization

Abstract

The distribution function ${P}_{L}(s)$ 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 $\ensuremath{\beta}$ and $\ensuremath{\nu}$ for $d=3$ are found as $\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01$, $\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities $n>1$.