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Journal of Physics A Mathematical and General

Finite-size effects on the critical structure factor of the two-dimensional Ising model

Norman C. Bartelt, T. L. Einstein

Published Jun 1, 1986• 20 citations
Publisher

Abstract

Explicit calculations of the critical structure factor of the planar Ising model on square lattices with a variety of boundaries are compared with the results of conformal invariance. Monte Carlo methods are used for square, circular and 2*1 rectangular geometries with free boundary conditions. Transfer matrices are used for an infinite strip with periodic boundary conditions. Good agreement is found for wavevectors less than a quarter of a reciprocal lattice spacing.

DOI
10.1088/0305-4470/19/8/023

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Explicit calculations of the critical structure factor of the planar Ising model on square lattices with a variety of boundaries are compared with the results of conformal invariance. Monte Carlo methods are used for square, circular and 2*1 rectangular geometries with free boundary conditions. Transfer matrices are used for an infinite strip with periodic boundary conditions. Good agreement is found for wavevectors less than a quarter of a reciprocal lattice spacing.
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The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al -2 , where l is the distance from the surface. The surface critical exponents of this system are non-universal. Under a conformal mapping onto a strip of width L, the Al -1 inhomogeneity transforms into A((L/ pi )sin( pi L/L)) -1 . For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit. The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance.
Anisotropy effects on the finite-size critical behavior of a two-dimensional Ising model on a general triangular lattice in an infinite-strip geometry with periodic, antiperiodic, and free boundary conditions (bc) in the finite direction are investigated. Exact results are obtained for the scaling functions of the finite-size contributions to the free energy density. With ξ(>) the largest and ξ(<) the smallest bulk correlation length at a given temperature near criticality, we find that the dependence of these functions on the ratio ξ(<)/ξ(>) and on the angle parametrizing the orientation of the correlation volume is of geometric nature. Since the scaling functions are independent of the particular microscopic realization of the anisotropy within the two-dimensional Ising model, our results provide a limited verification of universality. We explain our observations by considering finite-size scaling of free energy densities of general weakly anisotropic models on a d-dimensional film (i.e., in an L×∞(d-1) geometry) with bc in the finite direction that are invariant under a shear transformation relating the anisotropic and isotropic cases. This allows us to relate free energy scaling functions in the presence of an anisotropy to those of the corresponding isotropic system. We interpret our results as a simple and transparent case of anisotropic universality, where, compared to the isotropic case, scaling functions depend additionally on the shape and orientation of the correlation volume. We conjecture that this universality extends to cases where the geometry and/or the bc are not invariant under the shear transformation and argue in favor of validity of two-scale factor universality for weakly anisotropic systems.
Abstract We derive exact finite-size corrections for the free energy F of the Ising model on the square lattice with Brascamp–Kunz boundary conditions. We calculate ratios r p ( ρ ) of pth coefficients of F for the infinitely long cylinder ( ) and the infinitely long Brascamp–Kunz strip ( ) at varying values of the aspect ratio . Like previous studies have shown for the two-dimensional dimer model, the limiting values p → ∞ of r p ( ρ ) exhibit abrupt anomalous behavior at certain values of ρ. These critical values of ρ and the limiting values of the finite-size-expansion-coefficient ratios differ, however, between the two models.