Hopf symmetry-protected topological phase at the intersection of magnetic and spin loop-current order

Hopf terms are topological theta terms that are associated with a host of interesting physics, including anyons, statistical transmutation, chiral edge states, and the quantum spin-Hall effect. Here, we show that Hopf terms generically appear in two-dimensional metals without spin-orbit coupling at the intersection of magnetic and spin loop-current order. In the locally ordered, but globally disordered, phase the system is governed by the Hopf term and realizes a Hopf symmetry-protected topological phase. This phase is protected by the SU(2) spin rotation symmetry, is gapped in the bulk, has chiral gapless edge states, and its spin-Hall conductance is quantized. Lattice models that realize this phase are introduced. In addition, we provide an elementary proof that the $θ$ angle of the Hopf term must be quantized to multiples of $π$ in non-relativistic systems, thereby precluding anyonic skyrmions in condensed matter systems.