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physica status solidi (b)

Transfer‐matrix study of critical properties of the Ising model following from the three‐cluster MFRG approach

G. Kamieniarz, Daria M. Tomecka

Published Dec 1, 2005• 0 citations
Publisher

Abstract

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)

DOI
10.1002/pssb.200562469

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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)
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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.