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arXiv:cond-mat/9912090 · Journal of Physics A Mathematical and General

Finite-size scaling corrections in two-dimensional Ising and Potts ferromagnets

S L A de Queiroz

Published Jan 24, 2000• 27 citations
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Abstract

Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out.

DOI
10.1088/0305-4470/33/4/306

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Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out.
. Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out. 1. Introduction The systematic study of subdominant terms in scaling provides researchers with guidelines on how to extrapolate, e.g. finite-size results to the thermodynamic limit [1--5]. Also, it may in itself bring out connections to underlying physical properties. These latter may be universal, such as the relationship between critical free-energy finite-width correction and conforma...
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
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
Recently the author suggested studying the scaling properties of complex two-dimensional phase transitions by calculating the universal amplitudes of interfacial free energies on a torus geometry employing Monte Carlo simulations. He tests this method at the 3*3 phase transition in the Ising lattice gas on a triangular lattice belonging to the chiral three-state Potts universality class. The interfacial free energies for this lattice gas model can be written in terms of free energy differences between systems with different sizes. The algorithm to calculate these free energy differences is presented. Using lattice sizes up to 18*18 and modest computing effort, he finds that the universal amplitude A=1.220.02, in accordance with the exact analytical value for the non-chiral three-state Potts model, A=1.203291 . . ., which he determined earlier.
We investigate the finite-temperature corrections to scaling in the three-state square-lattice Potts antiferromagnet, close to the critical point at T=0. Numerical diagonalization of the transfer matrix on semi-infinite strips of width L sites, 4 < or = L < or = 14, yields finite-size estimates of the corresponding scaled gaps, which are extrapolated to L --> infinity. Owing to the characteristics of the quantities under study, we argue that the natural variable to consider is x identical with L e(-2 beta). For the extrapolated scaled gaps we show that square-root corrections, in the variable x, are present, and provide estimates for the numerical values of the amplitudes of the first- and second-order correction terms, for both the first and second scaled gaps. We also calculate the third scaled gap of the transfer matrix spectrum at T=0, and find an extrapolated value of the decay-of-correlations exponent, eta(3)=2.00(1). This is at odds with earlier predictions, to the effect that the third relevant operator in the problem would give eta(P(stagg))=3, corresponding to the staggered polarization.