The Gonihedric Ising Model and Glassiness

Abstract

(Pre-)History of the ModelThe Gonihedric 3D Ising model is a lattice spin model in which planar Peierls boundaries between + and − spins can be created at zero energy cost. Instead of weighting the area of Peierls boundaries as the case for the usual 3D Ising model with nearest neighbour interactions, the edges, or "bends" in an interface are weighted, a concept which is related to the intrinsic curvature of the boundaries in the continuum.The model is a generalised Ising model living on a cubic 3D lattice with nearest neighbour, next to nearest-neighbour and plaquette interactions. The ratio between the couplings of these three terms is fixed to a one parameter family which endows the model with unusual properties both in and out of equilibrium. Of particular interest for the discussion here will be that the model manifests all the indications of glassy behaviour without any recourse to quenched disorder, whilst still possessing a crystalline low temperature phase in equilibrium.In these notes we follow a roughly chronological order by first reviewing the background to the formulation of the model, before moving on to the elucidation of the equilibrium phase diagram by various means and then to the investigation of the non-equilibrium, glassy behaviour of the model. We apologize in advance for our narrow focus on things Gonihedric at the expense of other lattice models with glassy behaviour, since the aim is to concentrate on giving an overview of the Gonihedric Ising model in 3D.The model has an unusual genesis since it was originally introduced as a potential discretization of string theory. The Nambu-Goto [1] action (or Hamiltonian in statistical mechanical language) in bosonic string theory is given by the area swept out by the string worldsheet as it moves through spacetime. Directly discretizing euclideanized versions of this action produced ensembles of surfaces, i.e. string worldsheets, which were dominated by collapsed and irregular configurations such as that in Fig.1 and which were unsuitable for

Odd fracton theories, proximate orders, and parton constructions

“…The g → ∞ limit, corresponding to the confined phase of the X-cube model, leads to an ordered phase of the Ising spins in which all plaquette terms are minimized. While the precise nature of the ordering is subtle and to our knowledge not completely characterized [54][55][56], it seems likely that it will involve the spontaneous breaking of the subsystem symmetries that are characteristic of plaquette Ising models. We now wish to perform the same duality transformation for an odd X-cube gauge theory.…”

Plaquette Ising models, degeneracy and scaling

“…In general, outside the field of disordered systems Ising models with multispin interactions have been less explored [6,7], both because such interactions are less common in real materials and also because in many cases such models display first-order transitions. These might be regarded as a priori less promising subjects for numerical investigation, although the general scaling theory for first-order transitions, initiated in [8][9][10] and developed further in [11][12][13][14][15] and [16][17][18][19], is now generally well-understood.…”