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arXiv:1612.00060 · The European Physical Journal Special Topics

Plaquette Ising models, degeneracy and scaling

Desmond A. Johnston, Marco Mueller, Wolfhard Janke E

Published Apr 1, 2017• 40 citations
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Abstract

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.

DOI
10.1140/epjst/e2016-60329-4

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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.
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.
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 note that the standard inverse system volume scaling for finite-size corrections at a first-order phase transition (i.e., 1/L31/{L}^{3}1/L3 for an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ lattice in 3D) is transmuted to 1/L21/{L}^{2}1/L2 scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exemplar of just such a system. We use multicanonical simulations of this model to generate high-precision data which provide strong confirmation of the nonstandard finite-size scaling law. The dual to the gonihedric model, which is an anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also displays the nonstandard scaling.
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.
A well-known feature of first-order phase transitions is that fixed boundary conditions can strongly influence finite-size corrections, modifying the leading corrections for an L 3 lattice in 3 d from order 1/ L 3 under periodic boundary conditions to 1/ L . A rather similar effect, albeit of completely different origin, occurs when the system possesses an exponential low- temperature phase degeneracy of the form 2 3 L which causes for periodic boundary conditions a leading correction of order 1/ L 2 in 3 d . We discuss a 3 d plaquette Hamiltonian ("gonihedric") Ising model, which displays such a degeneracy and manifests the modified scaling behaviour. We also investigate an apparent discrepancy between the fixed and periodic boundary condition latent heats for the model when extrapolating to the thermodynamic limit.
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.