Spin-Charge Groups for Fermions in Fluids and Crystals: General Structures and Physical Consequences

Known symmetry groups are insufficient to describe the various couplings among spin, charge, and spatial degrees of freedom in fermionic systems. To address this problem, we introduce spin-charge groups (SCGs), which provide a unified framework for fermionic symmetries. SCGs incorporate spin and charge operations as internal' symmetries, spatial and temporal operations as external’ symmetries, together with their couplings and projective twists. After deriving the general group structure of SCGs, we explore their applications in concrete physical systems, including ${}^3$He superfluids, charge-4e superconductors, collinear magnets with spin-fluxes, and superconductors with coexisting magnetic orders. We show that SCGs can enforce additional band degeneracies, Chern numbers and cross spin-charge responses. Hence SCGs provide a symmetry-based route toward the classification and exploration of new phases of matter even when strong interactions are included.