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Physica A Statistical Mechanics and its Applications

Convergence of finite size scaling analysis: Weak size dependence of renormalization transformations

Mirosław R. Dudek, J. M. J. van Leeuwen, Henk W. J. Blöte

Published Nov 1, 1987• 6 citations
Publisher
DOI
10.1016/0378-4371(87)90114-2

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•Jan 1, 2001•3 citations
A theory of finite-size rescaling transformations (FSRT) is derived. It yields accurate estimates of the critical properties of statistical-mechanical systems of infinite extent. The arguments used are analogous to, but different from, conventional renormalization-group theory. The FSRT do not involve block spins and do not introduce any new couplings beyond those already in the problem. A technique which was suggested phenomenologically by Nightingale is derived as a special case of the FSRT theory. The optimization of convergence rates, and the relationship between FSRT and the finite-size scaling hypotheses of Fisher, are examined. Some new numerical results are presented.
An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent. PACS: 05.10.Cc, 64.60.Ak, 05.50.+q, 75.10.Hk 1 1