Interacting-cluster spin liquids with robust flat bands evolving into higher-rank half-moon phases and topological Lifshitz transitions

Classical spin liquids are disordered magnetic phases, governed by local constraints that often give rise to flat-band ground states. When constraints take the form of a zero-divergence field within a cluster of spins, the spin liquid is often described by an emergent Coulomb gauge theory. Here we introduce an interaction $\eta$ between these clusters of spins which compete with the zero-divergence field. Using a framework embracing both the connectivity matrices of graph theory and the topology of band structures, we develop a generic theory of interacting-cluster Hamiltonians. We show how flat bands remain at zero energy up to finite interaction $\eta$, until a dispersive band becomes negative, stabilizing a spiral spin liquid with a hypersurface of ground-state manifold in reciprocal space. This hypersurface can be interpreted as an effective Fermi surface in the spectrum of the parent system, acting as a tunable energy selector despite the absence of particle filling. This effective Fermi surface serves as a mold for the apparition of the half-moon patterns in the equal-time structure factor. Our generic approach enables to extend the notion of half moons to the perturbation of higher-rank Coulomb fields and pinch-line spin liquids. In particular, multi-fold half moons appear when unconventional gauge charges, such as potential fractons, are stabilized in the ground state. Finally, half-moon phases can be tuned across the equivalent of a Lifshitz transition, when the hypersurface manifold changes topology.