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arXiv:1601.03997 · Nuclear Physics B

Exact solutions to plaquette Ising models with free and periodic boundaries

Marco Mueller, Desmond A. Johnston, Wolfhard Janke E

Published Nov 11, 2016• 12 citations
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Abstract

An anisotropic limit of the 3d plaquette Ising model, in which the plaquette couplings in one direction were set to zero, was solved for free boundary conditions by Suzuki (1972) [1], who later dubbed it the fuki-nuke, or “no-ceiling”, model. Defining new spin variables as the product of nearest-neighbour spins transforms the Hamiltonian into that of a stack of (standard) 2d Ising models and reveals the planar nature of the magnetic order, which is also present in the fully isotropic 3d plaquette model. More recently, the solution of the fuki-nuke model was discussed for periodic boundary conditions, which require a different approach to defining the product spin transformation, by Castelnovo et al. (2010) [2]. We clarify the exact relation between partition functions with free and periodic boundary conditions expressed in terms of original and product spin variables for the 2d plaquette and 3d fuki-nuke models, noting that the differences are already present in the 1d Ising model. In addition, we solve the 2d plaquette Ising model with helical boundary conditions. The various exactly solved examples illustrate how correlations can be induced in finite systems as a consequence of the choice of boundary conditions.

DOI
10.1016/j.nuclphysb.2016.11.005

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An anisotropic limit of the 3d plaquette Ising model, in which the plaquette couplings in one direction were set to zero, was solved for free boundary conditions by Suzuki (1972) [1], who later dubbed it the fuki-nuke, or “no-ceiling”, model. Defining new spin variables as the product of nearest-neighbour spins transforms the Hamiltonian into that of a stack of (standard) 2d Ising models and reveals the planar nature of the magnetic order, which is also present in the fully isotropic 3d plaquette model. More recently, the solution of the fuki-nuke model was discussed for periodic boundary conditions, which require a different approach to defining the product spin transformation, by Castelnovo et al. (2010) [2]. We clarify the exact relation between partition functions with free and periodic boundary conditions expressed in terms of original and product spin variables for the 2d plaquette and 3d fuki-nuke models, noting that the differences are already present in the 1d Ising model. In addition, we solve the 2d plaquette Ising model with helical boundary conditions. The various exactly solved examples illustrate how correlations can be induced in finite systems as a consequence of the choice of boundary conditions.
A 3D Ising model with a purely plaquette, 4-spin interaction displays a planar flip symmetry intermediate between a global and a gauge symmetry and as a consequence has a highly degenerate low temperature phase and no standard magnetic order parameter. This plaquette Hamiltonian is a particular case of a family of 3D gonihedric Ising models defined by Savvidy and Wegner. An anisotropic variant of the purely plaquette gonihedric model, originally discussed as the ‘Fuki-Nuke’ model by Suzuki, is non-trivially equivalent to a stack of 2D Ising models, each of which can magnetize independently at the phase transition point. Consideration of this anisotropic model suggests that a suitable order parameter in the isotropic case may also be constructed using a form of planar magnetization, in which nearest neighbour correlators 〈σ i σ j 〉 summed over planes replace a sum over spin values 〈σ i 〉. We conduct Monte-Carlo simulations to investigate this and related candidate order parameters in a toy model of gonihedric ground states, the Fuki-Nuke model and the isotropic plaquette gonihedric model itself.
The purely plaquette 3D Ising Hamiltonian with the spins living at the vertices of a cubic lattice displays several interesting features. The symmetries of the model lead to a macroscopic degeneracy of the low-temperature phase and prevent the definition of a standard magnetic order parameter. Consideration of the strongly anisotropic limit of the model suggests that a layered, “fuki-nuke” order still exists and we confirm this with multi-canonical simulations. The macroscopic degeneracy of the low-temperature phase also changes the finite-size scaling corrections at the first-order transition in the model and we see this must be taken into account when analyzing our measurements.
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.
In this paper we conduct a careful multicanonical simulation of the isotropic 3d plaquette (“gonihedric”) Ising model and confirm that a planar, fuki-nuke type order characterises the low-temperature phase of the model. From consideration of the anisotropic limit of the model we define a class of order parameters which can distinguish the low- and high-temperature phases in both the anisotropic and isotropic cases. We also verify the recently voiced suspicion that the order parameter like behaviour of the standard magnetic susceptibility χm seen in previous Metropolis simulations was an artefact of the algorithm failing to explore the phase space of the macroscopically degenerate low-temperature phase. χm is therefore not a suitable order parameter for the model.
Partition function zeros are powerful tools in understanding critical behavior. In this paper, we present results of the Fisher zeros of two-dimensional Ising models, in the framework of free-fermion eight-vertex model. First, we succeed in finding special boundary conditions for the free-fermion model, under which the partition function of a finite lattice can be expressed in a double product form. Using appropriate mappings, these boundary conditions are transformed into the corresponding versions of the square, triangular, and honeycomb lattice Ising models. Each Ising model is studied in the cases of a zero field and of an imaginary field i ( π / 2 ) k B T . For the square lattice model, we rediscover the famous Brascamp-Kunz (B-K) boundary conditions. For the triangular and honeycomb lattice models, we obtain the B-K-type boundary conditions, and the Fisher zeros are conveniently solved from the product form of partition function. The advantage of B-K-type boundary conditions is that the Fisher zeros of any finite lattice exactly lie on certain loci, and the accumulation points of zeros can be easily determined in the thermodynamic limit. Our finding and method would be very helpful in studying the partition function zeros of vertex and Ising models.