Abstract Based on the results obtained in (Hucht 2017 J. Phys. A: Math. Theor. 50 065201), we show that the partition function of the anisotropic square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, is given by the determinant of an M/2 × M/2 Hankel matrix, that equivalently can be written as the Pfaffian of a skew-symmetric M × M Toeplitz matrix. The M − 1 independent matrix elements of both matrices are Fourier coefficients of a certain symbol function, which is given by the ratio of two characteristic polynomials. These polynomials are associated to the different directions of the system, encode the respective boundary conditions, and are directly related through the symmetry of the considered Ising model under exchange of the two directions. The results can be generalized to other boundary conditions and are well suited for the analysis of finite-size scaling functions in the critical scaling limit using Szegő’s theorem.
Abstract Based on the results obtained in (Hucht 2017 J. Phys. A: Math. Theor. 50 065201), we show that the partition function of the anisotropic square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, is given by the determinant of an M/2 × M/2 Hankel matrix, that equivalently can be written as the Pfaffian of a skew-symmetric M × M Toeplitz matrix. The M − 1 independent matrix elements of both matrices are Fourier coefficients of a certain symbol function, which is given by the ratio of two characteristic polynomials. These polynomials are associated to the different directions of the system, encode the respective boundary conditions, and are directly related through the symmetry of the considered Ising model under exchange of the two directions. The results can be generalized to other boundary conditions and are well suited for the analysis of finite-size scaling functions in the critical scaling limit using Szegő’s theorem.
Abstract The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size L × M and temperature. We start with the dimer method of Kasteleyn, McCoy and Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, F ( L , M ) = F s t r i p ( L , M ) + F s t r i p r e s ( L , M ) , where the residual part F s t r i p r e s ( L , M ) contains the nontrivial finite- L contributions for fixed M . It is given by the determinant of a M / 2 × M / 2 matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. While F s t r i p r e s ( L , M )
Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.
We study the boundary correlation function of the fixed-to-free boundary-condition-changing (bcc) operators in the square-lattice Ising model. First, we find a formula for representing a large class of two-point boundary correlation functions using a 2 × 2 block Toeplitz determinant. Using this formula the correlation function of the fixed-to-free bcc operator is represented using block Toeplitz determinants, for arbitrary, uniformly anisotropic couplings. This block Toeplitz determinant is transformed into a scalar Toeplitz determinant when the size of the matrix is an even number. We use Szegö’s theorem and the Fisher–Hartwig theorem to identify the asymptotic behavior of this scalar Toeplitz determinant.
Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.
We analyze the exact partition function of the anisotropic Ising model on finite M × N rectangular lattices under four different boundary conditions (periodic–periodic (pp), periodic–antiperiodic (pa), antiperiodic–periodic (ap) and antiperiodic–antiperiodic (aa)) obtained by Kaufman (1949 Phys. Rev. 76 1232), Wu and Hu (2002 J. Phys. A: Math. Gen. 35 5189) and Kastening (2002 Phys. Rev. E 66 057103)). We express the partition functions in terms of the partition functions Zα, β(J, k) with (α, β) = (0, 0), (1/2, 0), (0, 1/2) and (1/2, 1/2), J is an interaction coupling and k is an anisotropy parameter. Based on such expressions, we then extend the algorithm of Ivashkevich et al (2002 J. Phys. A: Math. Gen. 35 5543) to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. Our result is f = fbulk + ∑∞p = 0fp(ρ, k)S−p − 1, where f is the free energy of the system, fbulk is the free energy of the bulk, S = MN is the area of the lattice and ρ = M/N is the aspect ratio. All coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ρeff = ρ/sinh 2Jc and show that for pp and aa boundary conditions all finite size correction terms are invariant under the transformation ρeff → 1/ρeff.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.