Classical Spin Liquids as Condensates of Spinons

The classical 1D Ising model is a precursor for understanding fractionalization of spins into spinons. We present a method for constructing a wide variety of classical Ising spin liquids, in two and three dimensions, by starting with a network of decoupled 1D Ising models and condensing bound states of spinons. Each distinct pattern of spinon condensation leads to a different classical spin liquid of which the simplest pattern yields ice models, while more complex patterns lead to fracton spin liquids. A given spinon condensation pattern defines a Gauss law (constrainer) and generalized ring exchange process (fluctuator). Each model derived this way has a corresponding set of flat bands with a variety of nodal point and line touchings, characterized by a coarse grained tensor gauge theory. We characterize the multipolar nature of the excitations in terms of irreducible representations of the lattice symmetries. We illustrate with a host of examples on various lattices.