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arXiv:cond-mat/0607008 · Physical Review E

Eliminated corrections to scaling around a renormalization-group fixed point: Transfer-matrix simulation of an extended d=3 Ising model

Yoshihiro Nishiyama

Published Jul 26, 2006• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.74.016120

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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.
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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 of square geometry with periodic boundaries oriented either along <10> or along <11> direction 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 proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
Finite-size scaling at fixed renormalization-group invariant is a powerful and flexible technique to analyze Monte Carlo data at a critical point. It consists in fixing a given renormalization-group invariant quantity to a given value, thereby trading its statistical fluctuations with those of a parameter driving the transition. One remarkable feature is the observed significant improvement of statistical accuracy of various quantities, as compared to a standard analysis. We review the method, discussing in detail its implementation, the error analysis, and a previously introduced covariance-based optimization. Comprehensive benchmarks on the Ising model in two and three dimensions show large gains in the statistical accuracy, which are due to cross-correlations between observables. As an application, we compute an accurate estimate of the inverse critical temperature of the improved O(2) ϕ^{4} model on a three-dimensional cubic lattice.
Based on Novotny's transfer-matrix method, we simulated the (stacked) triangular Ising anti-ferromagnet embedded in the space with the dimensions variable in the range 2 < or = d < or = 3. Our aim is to investigate the criticality of the XY universality class for 2 < or = d < or = 3. For that purpose, we employed an extended version of the finite-size-scaling analysis developed by Novotny, who utilized this scheme to survey the Ising criticality (ferromagnet) for 1 < or = d < or = 3. Diagonalizing the transfer matrix for the system sizes N up to N = 17 , we calculated the d -dependent correlation-length critical exponent nu(d). Our simulation result nu(d) appears to interpolate smoothly the known two limiting cases, namely, the Kosterlitz-Thouless (KT) and d = 3 XY universality classes, and the intermediate behavior bears close resemblance to that of the analytical formula via the 1/N-expansion technique. Methodological details including the modifications specific to the present model are reported.
We have undertaken a large-scale computer simulation of the 3d Ising model using the Monte Carlo Renormalization Group (MCRG) method. Our calculation improves on past calculations in three ways. Firstly, larger lattices are used -- we have run on 128{sup 3} as well as 64{sup 3}. Secondly, many more spin operators -- 53 even and 46 odd -- are measured. Lastly, we have incorporated more efficient ways of spin update -- Wolff's single-cluster variant of the Swendsen and Wang algorithm. In addition to calculating the critical temperature the thermal and magnetic exponents, and improving the estimate of the correction-to-scaling exponent, we are aiming to verify the presence of redundant operators. We do not yet have sufficient statistics to give final results so here we describe our calculation and present preliminary results which demonstrate that we have control over both finite size effects and systematic errors arising from truncation. 16 refs., 7 figs.
Using transfer-matrix extended phenomenological renormalization-group methods [M.A.Yurishchev, Nucl. Phys. B (Proc. Suppl.) 83-84, 727 (2000); hep-lat/9908019; J. Exp. Theor. Phys. 91, 332 (2000); cond-mat/0108002] the improved estimates for the critical temperature of spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J⃗=(J′,J′,J){\vec J}=(J',J',J)J=(J′,J′,J) are obtained. Universality of both fundamental critical exponents yty_tyt​ and yhy_hyh​ is confirmed. We show also that the critical finite-size scaling amplitude ratios Aχ(4)Aκ/Aχ2A_{χ^{(4)}}A_κ/A_χ^2Aχ(4)​Aκ​/Aχ2​, Aκ′′/AχA_{κ^{''}}/A_χAκ′′​/Aχ​, and Aκ(4)/Aχ(4)A_{κ^{(4)}}/A_{χ^{(4)}}Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/JΔ=J'/JΔ=J′/J.