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arXiv:2303.03484 · Journal of Physics A Mathematical and Theoretical

Exact coefficients of finite-size corrections in the Ising model with Brascamp–Kunz boundary conditions and their relationships for strip and cylindrical geometries

N. Sh. Izmailian, Ralph Kenna B, V. V. Papoyan

Published Sep 13, 2023• 1 citation
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Abstract

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.

DOI
10.1088/1751-8121/acf96b

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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.
Finite-size scaling, finite-size corrections, and boundary effects for critical systems have attracted much attention in recent years. Here we derive exact finite-size corrections for the free energy F and the specific heat C of the critical ferromagnetic Ising model on the Mu x 2 Nu square lattice with Brascamp-Kunz (BK) boundary conditions [J. Math. Phys. 15, 66 (1974)] and compare such results with those under toroidal boundary conditions. When the ratio xi/2=(Mu+1)/2 Nu is smaller than 1 the behaviors of finite-size corrections for C are quite different for BK and toroidal boundary conditions; when ln(xi/2) is larger than 3, finite-size corrections for C in two boundary conditions approach the same values. In the limit Nu-->infinity we obtain the expansion of the free energy for infinitely long strip with BK boundary conditions. Our results are consistent with the conformal field theory prediction for the mixed boundary conditions by Cardy [Nucl. Phys. B 275, 200 (1986)] although the definitions of boundary conditions in two cases are different in one side of the long strip.
The Ising model in two dimensions with the special boundary conditions of Brascamp and Kunz is analyzed. Leading and subdominant scaling behavior of the Fisher zeros are determined exactly. The exact finite-size scaling, with corrections, of the specific heat is determined both at critical and effective critical (pseudocritical) points. The shift exponents associated with the scaling of these effective critical points are not the same as the inverse correlation length critical exponent. All corrections to scaling are analytic.
We study the finite-size scaling properties of the Ising model on the Möbius strip and the Klein bottle. The results are compared with those of the Ising model under different boundary conditions, that is, the free, cylindrical, and toroidal boundary conditions. The difference in the magnetization distribution function p(m) for various boundary conditions is discussed in terms of the number of the percolating clusters and the cluster size. We also find interesting aspect-ratio dependence of the value of the Binder parameter at T = T(c) for various boundary conditions. We discuss the relation to the finite-size correction calculations for the dimer statistics.
INTRODUCTION Finite-size scaling (FSS) is a well established technique for the extraction of critical exponents from finite volume analyses [1]. Such exponents characterise critical phenomena at a second-order phase transition. The simplest model exhibiting such a transition is the Ising model in two dimensions, which, despite a long history and extensive study, still o#ers new results and insights. Here, we study the model under the special boundary conditions of Brascamp and Kunz [2] to extract new information and to help resolve some hitherto puzzling features of FSS. Let CL (#) be the specific heat at inverse temperature # for a system of linear extent L. FSS of the specific heat is characterized by the location of its peak, #L , its height CL (# L ) and its value at the infinite-volume critical point CL (# c ). The peak position, #L , is a pseudocritical point which typically approaches # c as L ## as |# # c L -# , (1) where # is the shift exponent. In two dimensions,
We study the (analytic) finite-size corrections in the Ising model on the strip with free, fixed (++), and mixed boundary conditions. For fixed (++) boundary conditions, the spins are fixed to the same values on two sides of the strip. We find that subdominant finite-size corrections to scaling should be to the form a(p)/N(2p-1) for the free energy f(N) and b(p)/N(2p-1) for inverse correlation length ξ(N)(-1), with integer value of p. We investigate the set {a(p),b(p)} by exact evaluation and their changes upon varying anisotropy of coupling. We find that the amplitude ratios b(p)/a(p) remain constant upon varying coupling anisotropy. Such universal behavior is correctly reproduced by the conformal perturbative approach.
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.