Classification of classical spin liquids: Topological quantum chemistry and crystalline symmetry

Classical spin liquids (CSLs) can be classified by their lowest-energy flat bands. This work advances the classification scheme by integrating crystalline symmetry and topological quantum chemistry into it. The authors present here a mathematical framework to analyze the band representations of flat bands in CSLs, illuminating how symmetry protects different classes of CSLs. Concrete examples, such as the construction of a symmetry-protected pinch-line algebraic CSL, demonstrate the power of this new classification approach.