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Classical spin liquids have been a very active subject of research in thelast few years. A very rich variety of cases have been shown to exist,including short-range and algebraic spin liquids displaying dipolarcorrelations at zero temperature. In this article, we investigate differentfamilies of classical spins liquids by combining analytical techniques andMonte Carlo simulations. Our study relies on the Luttinger-Tisza approximation(LTA), which is associated with the constraint vector function in momentumspace, whose general properties allow for a classification of different spinliquids. We show that the general properties of the LTA provide a framework foridentifying and accurately characterizing the different types of spin liquidsin different geometries. We apply our approach to three different families ofspin liquids defined on the checkerboard and kagome lattices, which exhibit aremarkable range of situations, including various cases of algebraic andshort-range spin liquids. Additionally, we analyze the effective Gauss lawemerging from different kinds of spin liquids and identify states that exhibitadditional degeneracy lines. The presence of spin-liquid phases and pinch-pointsingularities are confirmed by Monte Carlo simulations validating our approach.Our study opens up avenues of research in the study of spin liquids, exploringalgebraic spin liquids with higher-rank gauge fields and as critical pointsdividing different types of classical spin liquids.

Combined approach to analyze and classify families of classical spin liquids