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arXiv:1611.05622 · Journal of Statistical Mechanics Theory and Experiment

Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios

Hendrik Hobrecht, Alfred Hucht

Published Feb 1, 2017• 15 citations
arXivHTMLPDF

Abstract

Abstract We present a systematic method to calculate the universal scaling functions for the critical Casimir force and the according potential of the two-dimensional Ising model with various boundary conditions. Therefore we start with the dimer representation of the corresponding partition function Z on an L × M square lattice, wrapped around a torus with aspect ratio ρ = L / M . By assuming periodic boundary conditions and translational invariance in at least one direction, we systematically reduce the problem to a 2 × 2 transfer matrix representation. For the torus we first reproduce the results by Kaufman and then give a detailed calculation of the scaling functions. Afterwards we present the calculation for the cylinder with open boundary conditions. All scaling functions are given in form of combinations of infinite products and integrals. Our results reproduce the known scaling functions in the limit of thin films ρ → 0 . Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.

DOI
10.1088/1742-5468/aa5280

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Abstract We present a systematic method to calculate the universal scaling functions for the critical Casimir force and the according potential of the two-dimensional Ising model with various boundary conditions. Therefore we start with the dimer representation of the corresponding partition function Z on an L × M square lattice, wrapped around a torus with aspect ratio ρ = L / M . By assuming periodic boundary conditions and translational invariance in at least one direction, we systematically reduce the problem to a 2 × 2 transfer matrix representation. For the torus we first reproduce the results by Kaufman and then give a detailed calculation of the scaling functions. Afterwards we present the calculation for the cylinder with open boundary conditions. All scaling functions are given in form of combinations of infinite products and integrals. Our results reproduce the known scaling functions in the limit of thin films ρ → 0 . Additionally, for the cylinder at criticality our results confirm the predictions from conformal field theory.
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 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.
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.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form a k / L x 2 k − 1 for the free energy f and b k ( p ) / L x 2 k − 1 and c k ( p ) / L x 2 k − 1 for inverse correlation lengths ξ p − 1 and ξ L − p − 1 , respectively, with integer values of k . By exact evaluation we find that the amplitude ratios b k ( p ) / a k and c k ( p ) / a k are universal and verify this universal behavior using a perturbative conformal approach.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form ak/Lx2k−1a_k/L^{2k-1}_xak​/Lx2k−1​ for the free energy f and bk(p)/Lx2k−1b_k(p)/L_x^{2k-1}bk​(p)/Lx2k−1​ and ck(p)/Lx2k−1c_k(p)/L_x^{2k-1}ck​(p)/Lx2k−1​ for inverse correlation lengths xip−1\\xi^{-1}_pxip−1​ and xiL−p−1\\xi^{-1}_{L-p}xiL−p−1​ , respectively, with integer values of k. By exact evaluation we find that the amplitude ratios bk(p)/akb_k(p)/a_kbk​(p)/ak​ and ck(p)/akc_k(p)/a_kck​(p)/ak​ are universal and verify this universal behavior using a perturbative conformal approach.
The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.