Hopf symmetry protected topological phases in the vicinity of spin orders
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914511159558144 |
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| author | Palle, Grgur |
| author_facet | Palle, Grgur |
| contents | 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 $θ$ angle of the Hopf term must be quantized to multiples of $π$ in non-relativistic systems, thereby precluding anyonic skyrmions in condensed matter systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hopf symmetry protected topological phases in the vicinity of spin orders Palle, Grgur Strongly Correlated Electrons 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 $θ$ angle of the Hopf term must be quantized to multiples of $π$ in non-relativistic systems, thereby precluding anyonic skyrmions in condensed matter systems. |
| title | Hopf symmetry protected topological phases in the vicinity of spin orders |
| topic | Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2510.01738 |