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arXiv:1407.6919 · Journal of the Korean Physical Society

Low-temperature behavior of the finite-size one-dimensional Ising model and the partition function zeros

Julian Lee

Published Sep 1, 2014• 10 citations
arXivHTMLPDF

Abstract

In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.

DOI
10.3938/jkps.65.676

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In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.
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Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.
The critical properties of an infinitely long Ising strip with finite width L joined periodically or antiperiodically are investigated by analyzing the distribution of partition function zeros. For periodic boundary condition, the the leading finite-size scaling of partition function zeros and its corrections are given. For antiperiodic boundary condition, the critical point of 2D Ising transition is one of the loci of the zeros, and the associated non-analyticity is identified as a first-order phase transition. The exact amount of the latent heat released by the transition is 4/L.
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 three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.