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

Sladequeiroz Instituto De, Potts Ferromagnets

Published Nov 21, 2007• 24 citations
OpenAlex

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. 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...

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

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arXiv:1105.6248•Physical Review E•Sep 6, 2011•2 citations
. 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...
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.
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.
Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/kB{k}_{B}kB​T as T\ensuremath{\rightarrow}Tc{T}_{c}Tc​. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.
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 consider strips of Ising spins at criticality. For strips of width N sites, subdominant (additive) finite-size corrections to scaling are assumed to be of the form a(k)/N(k) for the free energy, and b(k)/N(k) for inverse correlation length, with integer values of k. We investigate the set {a(k),b(k)} (k≥2) by exact evaluation and numerical transfer-matrix diagonalization techniques, and their changes upon varying anisotropy of couplings, spin quantum number S, and (finite) interaction range, in all cases for both periodic (PBCs) and free (FBCs) boundary conditions across the strip. We find that the coefficient ratios b(k)/a(k) remain constant upon varying coupling anisotropy for S=1/2 and first-neighbor couplings, for both PBCs and FBCs (albeit at distinct values in either case). Such apparently universal behavior is not maintained upon changes in S or interaction range.