Classification of classical spin liquids in the Large-N Limit*

Classical spin liquids (CSL) are arguably one of the most interesting types of classical matter, offering a canvas from which emergent quantum phases can emerge.  Described by the classical limit of gauge theories (electrostatics), these CSL 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 CSL in the limit of the large number (N) of spin components,  which allows them to be treated as real-valued scalars. We found that the ground state degeneracy is encoded in the flat bands at the bottom of the spectrum, with the classical spin liquids classified by the properties of these bands. Two categories emerge: (i) when there is a singular band touching between the top bands and the bottom ones, the system has algebraic correlations; 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 (ii) is when the flat bands are gapped from the top ones. In this case, the correlations are short-ranged, however, 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.