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arXiv:cond-mat/0204217 · Journal of Physics A Mathematical and General

Exact partition functions of the Ising model on M $times$ N planar lattices with periodic$ndash$aperiodic boundary conditions

Ming‐Chya Wu, Chin‐Kun Hu

Published Jun 14, 2002• 32 citations
arXivHTMLPDF

Abstract

The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on M × N square (sq), plane triangular (pt) and honeycomb (hc) lattices with periodic–periodic (pp), periodic–antiperiodic (pa), antiperiodic–periodic (ap) and antiperiodic–antiperiodic (aa) boundary conditions. The partition functions are used to calculate and plot the specific heat, C / k B , as a function of temperature, θ = k B T / J . We find that for the N × N sq lattice, C / k B for pa and ap boundary conditions are different from those for aa boundary conditions, but for the N × N pt and hc lattices, C / k B for ap, pa and aa boundary conditions have the same values. Our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite-size corrections of the Ising model.

DOI
10.1088/0305-4470/35/25/304

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Exact partition functions of the Ising model on M timestimestimes N planar lattices with periodicndashndashndashaperiodic boundary conditions
The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on M × N square (sq), plane triangular (pt) and honeycomb (hc) lattices with periodic–periodic (pp), periodic–antiperiodic (pa), antiperiodic–periodic (ap) and antiperiodic–antiperiodic (aa) boundary conditions. The partition functions are used to calculate and plot the specific heat, C / k B , as a function of temperature, θ = k B T / J . We find that for the N × N sq lattice, C / k B for pa and ap boundary conditions are different from those for aa boundary conditions, but for the N × N pt and hc lattices, C / k B for ap, pa and aa boundary conditions have the same values. Our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite-size corrections of the 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.
The exact partition functions of the Ising model on L × L square lattices with free boundary conditions are evaluated up to L = 22 for the first time by using a microcanonical transfer matrix and counting all possible spin states. The total number of states is 2 22×22 = 2 484 ≈ 5.0 × 10 145 for L = 22. Using the partition function zeros in the complex temperature plane of the Ising model on L × L square lattices with free boundary conditions, we analyze the critical behavior of the square-lattice Ising antiferromagnet and compare them with the results for periodic boundary conditions (the most popular boundary conditions). Our results show clearly that free boundary conditions are much more efficient than periodic boundary conditions in the study of an antiferromagnet.
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.
We analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is lambda=1 for all values of the helicity factor d . We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.