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Hopf symmetry protected topological phases in the vicinity of spin orders

Grgur Palle*

  • *Contact author: gpalle@illinois.edu
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Phys. Rev. B 113, 155155 – Published 27 April, 2026

DOI: https://doi.org/10.1103/szlj-7j6m

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 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 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 nonrelativistic systems, thereby precluding anyonic skyrmions in condensed matter systems.

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Phys. Rev. B 113, 155155– Published 27 April, 2026
  • Received 2 October 2025
  • Revised 3 February 2026
  • Accepted 31 March 2026
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