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Journal of Physics A Mathematical and General

Phenomenological renormalisation of Monte Carlo data

Michael N. Barber, Walter Selke

Published Nov 1, 1982• 62 citations
Publisher

Abstract

A method, akin to phenomenological renormalisation, for analysing Monte Carlo data in the critical region is proposed. The method is illustrated by an analysis of the structure factor of the two-dimensional axial next-nearest neighbour Ising model.
DOI
10.1088/0305-4470/15/11/009

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The accuracy of a phenomenological renormalisation which is based on Monte Carlo data is tested by investigating site percolation in a simple cubic lattice. The method appears to be very accurate and can yield precise estimates of the quantities of interest with small to moderate lattice sizes. The site percolation threshold of the lattice is predicted to be 0.3115+or-0.0005. If nu is the exponent of correlation length, the author finds beta / nu approximately=0.48+or-0.01 for the exponent of percolation probability and beta B / nu approximately=1.1+or-0.03 for the backbone exponent, in agreement with most accurate data currently available. An analysis of the data at the percolation threshold based on the standard finite-size scaling method also supports the results.
The modern techniques of field theory applied to critical phenomena, are briefly discussed, with particular emphasis on the use of the e-expansion to extract the asymptotic behaviour in the critical region. The dimensional regularisation and minimal subtraction method of renormalisation, Introduced by 't Hooft and Veltman, is outlined. These methods, along with a Monte Carlo simulation on a highly parallel computer, are utilised in extracting information pertaining to the behaviour near criticality of the random, site-diluted Ising model. The universal ratio of the susceptibility amplitude above and below the critical temperature is determined to one higher order than previously in an ɛ-expansion, within the framework of the replica formalism. The calculation is performed using a Taylor expansion, in the number of replicas, of the transverse propagator appearing In the Hamiltonian, about the longitudinal. This higher order correction goes in the right direction for agreement with experiment. An extension of the formalism to embrace the presence of two sets of replica spins is used to derive the o(ɛ¹/²) term in an expansion of the additional static - correlation function, C⁽ˢ⁾(q), arising from the non-interchangeability of the thermal and configurational averages In the random model. The appearance of C(sl(q) in the structure factor Implies that results of neutron scattering experiments to measure the susceptibility amplitude ratio must be re-Interpreted. Finally, analysis of the properties of block spin variables In a Monte Carlo simulation Is used to determine the phase diagram and critical exponents In two-dimensions. Possible Interpretations of the observed evolution of the exponent ratio, ß/υ, under Increasing dilution are discussed.