Decorated cluster lattices: A natural framework for classical spin liquids and flat bands

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. Importantly, decorated cluster lattices constitute an ideal playground for producing multiple flat bands-not only at the spectral bottom but also embedded within the spectrum. This proliferation of flat bands can be understood either from general connectivity arguments or, equivalently, by explicitly constructing compact localized eigenmodes in an associated hopping description.