Building cluster systems

Classical spin liquids are frustrated magnetic phases characterized by local constraints, flat bands in reciprocal space, and emergent gauge structures with distinctive signatures such as pinch points. These arise generally in \emph{cluster systems}, where spin interactions can be expressed as constraints on clusters of spins. In this work we present the different generic rules allowing to build such cluster systems together with a few tools allowing to quickly characterize it. We show that based on these rules, it is possible to conceive a tunable recipe for generating such models by decorating a parent lattice on its bonds and/or vertices with symmetry-compatible clusters. This approach highlights a key design trade-off: using fewer cluster types increases the number of flat bands and enhances spin-liquid behavior, but produces denser connectivity that is harder to realize experimentally. The framework is highly tunable, extends naturally to two and three dimensions, and provides a versatile toolbox for engineering new classical spin-liquid candidates with targeted features such as higher-rank pinch points or pinch lines. Among the catalog of cluster systems generated following the decoration scheme, a common feature appears to be the existence of additional flat bands in the spectrum, which motivates a detailed study of the origin of these flat bands and their associated physics.