Classical spin liquids are arguably one of the most interesting types of classical matter. They are described often by the classical limit of gauge theories (electrostatics) and can be upgraded to topological orders or gapless quantum liquid states if equipped with proper quantum dynamics. In this work, we present a classification scheme for classical spin liquids in the large-S limit. In this limit, the spin components are effectively real scalars, so the spectrum of the Hamiltonian accurately describes its properties. We found that the ground state degeneracy corresponds to flat bands at the bottom of the spectrum, and the flat band’s structure crucially decides the physics of the classical spin liquid. When there is a singular band-touching between the top bands and the bottom flat ones, the system has algebraic correlation. The ground state is then described by a generalized Gauss’s law, whose algebraic form is determined by the band touching structure. A much less studied category is when the flat bands are gapped from the top ones. In this case, the correlation is short-ranged, but the classical spin liquid can be distinguished from a trivial paramagnet by the fragile topological homotopy of the bottom bands. Besides building the general mathematical framework of the classification, I will also show some concrete examples and discuss experimental applications.

Please access here for the registration.