A characteristic feature of the 3 d plaquette Ising model is its planar subsystem symmetry. The quantum version of this model has been shown to be related via a duality to the X-Cube model, which has been paradigmatic in the new and rapidly developing field of fractons. The relation between the 3 d plaquette Ising and the X-Cube model is similar to that between the 2 d quantum transverse spin Ising model and the Toric Code. Gauging the global symmetry in the case of the 2 d Ising model and considering the gauge invariant sector of the high temperature phase leads to the Toric Code, whereas gauging the subsystem symmetry of the 3 d quantum transverse spin plaquette Ising model leads to the X-Cube model. A non-standard dual formulation of the 3 d plaquette Ising model which utilises three flavours of spins has recently been discussed in the context of dualising the fracton-free sector of the X-Cube model. In this paper we investigate the classical spin version of this non-standard dual Hamiltonian and discuss its properties in relation to the more familiar Ashkin–Teller-like dual and further related dual formulations involving both link and vertex spins and non-Ising spins.
Noisy Toric code and random-bond Ising model: The error threshold in a dual picture
