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Acta Physica Polonica A

Numerical Test of Molecular-Field Renormalization Group

G. Kamieniarz

Published Aug 1, 1997• 0 citations
Publisher

Abstract

A new transfer matrix approach has been worked out to test the predictions of the molecular-field renormalization group for square Ising clusters with a linear size up to L < 11.The convergence of the finite-size critical couplings towards the exact value for the molecular-field renormalization group is shown and the limit of the ratio yh/yt consistent with the corresponding universal value is revealed.

DOI
10.12693/aphyspola.92.423

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A new transfer matrix approach has been worked out to test the predictions of the molecular-field renormalization group for square Ising clusters with a linear size up to L < 11.The convergence of the finite-size critical couplings towards the exact value for the molecular-field renormalization group is shown and the limit of the ratio yh/yt consistent with the corresponding universal value is revealed.
Renormalization group recursions based on mean-field approximations [J. O. Indekeu, A. Maritan, and A. L. Stella, J. Phys.A15, L291 (1982)], commonly referred to as mean-field renormalization group methods (MFRG), have proven to be efficient and easily applicable for computing non-classical critical properties of lattice models. We give a fairly complete bibliography of applications to date, and extend previous test calculations of bulk, surface, and corner critical exponents in the two-dimensional Ising model to larger cluster sizes on triangular, square (including crossing bonds), and honeycomb lattices. Without much effort the exact value of the critical exponent ratioyH/yT is reproduced systematically with a precision of 2%. This ratio turns out to be the most accurate probe of non-classical critical behaviour that is available in the MFRG method.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
Abstract Using a transfer matrix technique and the Ising model, the predictions of the MFRG concept for clusters with linear size up to 21 have been tested for the three‐cluster MFRG approach. Even for small sizes of the clusters, the three‐cluster estimates of critical couplings and bulk critical exponents give the accuracy level equal to that of two‐cluster renormalization for much greater sizes. Performing the asymptotic analysis for the new data, the convergence of the surface critical exponent yhs towards the exact value is illustrated for the first time. Our improved method has enabled us to obtain the results for clusters with substantially greater sizes, to accelerate the calculations and also to confirm the reliability of the MFRG approach for the Ising model. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Extending the parameter space of the three-dimensional (d=3) Ising model, we search for a regime of eliminated corrections to finite-size scaling. For that purpose, we consider a real-space renormalization group (RSRG) with respect to a couple of clusters simulated with the transfer-matrix (TM) method. Imposing a criterion of "scale invariance," we determine a location of the nontrivial RSRG fixed point. Subsequent large-scale TM simulation around the fixed point reveals eliminated corrections to finite-size scaling. As anticipated, such an elimination of corrections admits systematic finite-size-scaling analysis. We obtained the estimates for the critical indices as nu=0.6245(28) and y(h)=2.4709(73). As demonstrated, with the aid of the preliminary RSRG survey, the transfer-matrix simulation provides rather reliable information on criticality even for d=3, where the tractable system size is restricted severely.