Hopf symmetry protected topological phases in the vicinity of spin orders

Abstract

Hopf terms are topological theta terms that are associated with a host of interesting physics, including anyons, statistical transmutation, chiral edge states, and the spin quantum Hall effect. Here, we show that Hopf terms can appear in two-dimensional metals without spin-orbit coupling in the vicinity of spin-ordered phases. In their vicinity, their spin-like order parameters have a finite amplitude, but fluctuating orientation. When both a magnetic and a spin loop-current order parameter fluctuate in the system, we show that the phase is governed by the Hopf term and realizes a Hopf symmetry protected topological phase. This phase is protected by the unbroken $\mathrm{SU}(2)$ 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 $\theta$ angle of the Hopf term must be quantized to multiples of $\pi$ in non-relativistic systems, thereby precluding anyonic skyrmions in condensed matter systems.