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arXiv:1507.05784 · Modern Physics Letters B

Macroscopic degeneracy and order in the 3D plaquette Ising model

Desmond A. Johnston, Marco Mueller, Wolfhard Janke E

Published Jul 14, 2015• 2 citations
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Abstract

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.

DOI
10.1142/s0217984915501092

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•Jan 1, 2014•0 citations
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.
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.
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.
It is known that fixed boundary conditions modify the leading finite-size corrections for an L3 lattice in 3d at a first-order phase transition from 1/L3 to 1/L. We note that an exponential low-temperature phase degeneracy of the form 23L will lead to a different leading correction of order 1/L2. A 3d gonihedric Ising model with a four-spin interaction, plaquette Hamiltonian displays such a degeneracy and we confirm the modified scaling behaviour using high-precision multicanonical simulations. We remark that other models such as the Ising antiferromagnet on the FCC lattice, in which the number of “true” low-temperature phases is not macroscopically large but which possess an exponentially degenerate number of low lying states may display an ef- fective version of the modified scaling law for the range of lattice sizes accessible in simulations.
We review some recent investigations of the 3d plaquette Ising model. This displays a strong first-order phase transition with unusual scaling properties due to the size-dependent degeneracy of the low-temperature phase. In particular, the leading scaling correction is modified from the usual inverse volume behaviour ∝ 1/L 3 to 1/L 2. The degeneracy also has implications for the magnetic order in the model which has an intermediate nature between local and global order and gives rise to novel fracton topological defects in a related quantum Hamiltonian.
The gonihedric Ising Hamiltonians defined in three and higher dimensions by Savvidy and Wegner provide an extensive, and little explored, catalogue of spin models on (hyper)cubic lattices with many interesting features. In three dimensions the kappa=0 gonihedric Ising model on a cubic lattice has been shown to possess a degenerate low-temperature phase and a first order phase transition, as well as interesting dynamical properties. The dual Hamiltonian to this may be written as an anisotropic Ashkin-Teller model and also has a degenerate low-temperature phase as a result of similar symmetries to the original plaquette action. It is possible to write an alternative dual formulation which utilizes three flavours of spins, rather than the two of the Ashkin-Teller model. This still possesses anisotropic couplings, but all the interaction terms are now four spin couplings and it acquires an additional, local gauge symmetry. We investigate this alternative dual Hamiltonian using zero temperature and mean-field methods together with Monte-Carlo simulations and discuss its properties and the relation to the Ashkin-Teller variant.
It is known that fixed boundary conditions modify the leading finite-size corrections for an L 3 lattice in 3d at a first-order phase transition from 1/L 3 to 1/L. We note that an exponential low-temperature phase degeneracy of the form 2 3L will lead to a different leading correction of order 1/L 2 .A 3 d gonihedric Ising model with a four-spin interaction, plaquette Hamiltonian displays such a de generacy and we confirm the modified scaling behaviour using high-precision multicanonical simulations. We remark that other models such as the Ising antiferromagnet on the FCC lattice, in which the number of “true” low-temperature phases is not macroscopically large but which possess an exponentially degenerate number of low lying states may display an effective version of the modified scaling law for the range of lattice sizes accessible in simulations.