Monte Carlo realization of Kadanoff block transformation in the 2d plane-rotator model

Abstract

Abstract The method combining Monte Carlo (MC) simulation and renormalization-group analysis has been carried out by several workers, and has turned out to be powerful in the study of critical phenomena. We propose a quite new scheme for realizing the Kadanoff block transformation by using MC simulation. The essential feature is to use a random selection of a spin out of each block to assign the block spin in conformity to the random nature of the MC simulation. We consider a system of N × N spins on a square lattice, and divide it into blocks of side l so that l × l spins are contained in a block; l = 1, 2, 4, …, [ N /2]. We then perform the standard MC simulation at a temperature T . After the system has come to equilibrium, the simulation provides a sequence of configurations from which useful information can be extracted. At each configuration, we assign block spins by picking a spin at random out of each block. We thus obtain a sequence of configurations of block spins from which correlation functions between block spins, g ( l, T ), can be calculated. These correlation functions reflect the transformation property of the Kadanoff blocks, and provide a convenient basis for the renormalization group analysis. This program was carried out in the 2d plane-rotator model. The calculation was made for square arrays of 1136(34 × 34) and 4536(66 × 66) spins with periodic boundary conditions. The experiment indicated that the correlation functions have four different temperature regions; I: 0 ⩽ T T T T . (Temperature is defined in units of the transition temperature predicted from the mean-field theory.) The correlation functions show power-law decay in I and II. The decay factor is in agreement with the spin-wave prediction in I, but deviates slightly in II. In III the correlation functions show a singular behavior. In IV the correlation functions show exponential decay. The correlation length derived from them has a singular temperature dependence as predicted by Kosterlitz, and becomes infinite at 0.47. The specific heat exhibits a peak at 0.51. It is pointed out that all these results are closely related to the behavior of vortices.