Plaquette Ising models, degeneracy and scaling

TL;DR

The paper analyzes the 3d plaquette Ising model, showing a strong first-order transition with nonstandard finite-size scaling caused by a macroscopic, size-dependent degeneracy $q=2^{3L}$. It develops a two-phase scaling framework, introduces a hybrid $2d/3d$ order parameter via fuki-nuke-like constructions, and validates scaling with multicanonical simulations that locate $\beta^{\infty}$ and interface tensions. In the quantum regime, the planar flip symmetry leads to a dual fracton topological order with sub-extensive degeneracy and constrained excitations, linking the classical degeneracy to fracton physics. The work highlights how subsystem symmetries alter scaling at first-order transitions and informs the design of order parameters and numerical strategies for degenerate systems, with broader implications for topological quantum codes.

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.

Figures (7)

• Figure 1: Flipping the value of the Ising spins on a face of a single cube does not change its contribution to the energy at $T=0$. All of the configurations shown have the same energy.
• Figure 2: A typical ground state of the $3d$ plaquette Hamiltonian showing the edges of the planes of spins that are flipped with respect to a purely ferromagnetic ground state dotted. Since any plane of spins in any of the three possible orientations may be flipped in such a configuration, the ground-state degeneracy on an $L^3$ lattice is $q=2^{3L}$.
• Figure 3: Some possible ground-state spin configurations for the dual model on a cube with flipped spins. The $\sigma,\tau$ values are shown at each site, as are the directions of the anisotropic interactions between the spins. Once again, the entire lattice may be tiled by compatible combinations of such cubes.
• Figure 4: Plot of the goodness-of-fit parameter $Q$ for fits on the extremal locations of the specific heat, $\beta^{C_V^{\rm max}}$, and Binder's energy cumulant, $\beta^{B^{\rm min}}$, of the $3d$ plaquette Ising model for different fitting ranges $L_{\rm min}$ -- $L_{\rm max}$. Green regions indicate acceptable fits. Upper row: Standard $1/L^3$ finite-size scaling ansatz. Lower row: $1/L^2$ finite-size scaling.
• Figure 5: The fuki-nuke parameters $m^{x}_{\rm abs}$ and $m^{x}_{\rm sq}$ (upper row) along with their respective susceptibilities $\chi$ normalised by the system volume (lower row) over shifted inverse temperature $\beta$ for several lattice sizes $L$.