Symmetry, Superposition and Fragmentation in Classical Spin Liquids: A General Framework and Applications to Square Kagome Magnets

Classical magnets exhibit exotic ground state properties such as spin liquids and fractionalization, promising a manifestation of superposition and projective symmetry construction in classical theory. While system-specific spin-ice or soft-spin models exist, a formal theory for general classical magnets remains elusive. Here, we introduce a generic symmetry group construction built from a vector field in a plaquette of classical spins, demonstrating how classical spins superpose in irreducible representations (irreps) of the symmetry group. The corresponding probability amplitudes serve as order parameters and local spins as fragmented excitations. The formalism offers a many-body vector field representation of diverse ground states, including spin liquids and fragmented phases described as degenerate ensembles of irreps. We apply the theory to a frustrated square Kagome lattice, where spin-ice or soft spin rules are inapt, to describe spin liquids and fragmented phases, all validated through irreps ensembles and unbiased Monte Carlo simulation. Our generic theory sheds light on previously unknown aspects of spin-liquid phases and fragmentation and broadens their applications to other branches of field theory.