Hopf symmetry protected topological phases in the vicinity of spin orders

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 spinlike 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>SU</a:mi> <a:mo>(</a:mo> <a:mn>2</a:mn> <a:mo>)</a:mo> </a:mrow> </a:math> 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>θ</b:mi> </b:math> angle of the Hopf term must be quantized to multiples of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>π</c:mi> </c:math> in nonrelativistic systems, thereby precluding anyonic skyrmions in condensed matter systems.