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Physical Review Letters

Monte Carlo Renormalization-Group Transformations in Momentum Space

Robert H. Swendsen

Published Oct 19, 1981• 13 citations
Publisher

Abstract

Methods of extending Monte Carlo renormalization-group calculations to include transformations in momentum space are proposed to investigate systems for which block spins are difficult or impossible to construct---such systems include fluids, for which particles are not confined to a lattice, three-dimensional fcc antiferromagnets, and lattice gauge models in four dimensions. The feasibility of these transformations is demonstrated by calculations in two and three dimensions.

DOI
10.1103/physrevlett.47.1159

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Methods of extending Monte Carlo renormalization-group calculations to include transformations in momentum space are proposed to investigate systems for which block spins are difficult or impossible to construct---such systems include fluids, for which particles are not confined to a lattice, three-dimensional fcc antiferromagnets, and lattice gauge models in four dimensions. The feasibility of these transformations is demonstrated by calculations in two and three dimensions.
We implement a momentum-space renormalization-group procedure into the Monte Carlo renormalization-group (MCRG) method. Usually the symmetries of the system must be known and incorporated into the renormalization group, but here we let the renormalization group be given by the Monte Carlo simulation itself. This allows the MCRG method to be applicable to systems where the underlying symmetries may not be known, such as in Monte Carlo studies of quantum systems. As a test of this method, we apply it to the study of the two-dimensional square-lattice q-state Potts model, both for integer values of q and for noninteger values of q near q=4.
A general method for optimizing real-space renormalization-group transformations is presented. It is applicable to both spin systems and lattice gauge theories, with use of Monte Carlo computer simulations. It provides substantial improvement in the convergence of critical exponents for the d=3d=3d=3 Ising model.
We present a real-space renormalization-group (RG) study of the nnn-component classical Heisenberg model in the large-nnn limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.
Renormalization-group trajectories produced by various real-space, block-spin transformations of the d=2d=2d=2 Ising model are studied, using a recently developed technique for calculating renormalized coupling parameters from a Monte Carlo simulation of a single system.