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.

arXiv comment17 pages, 5 figures

References in corpus (33)

• Anomalous Hall effect
• Classification of topological quantum matter with symmetries
• Advances in the Physics of Magnetic Skyrmions and Perspective for Technology
• Magnetic 2D materials and heterostructures
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Spontaneous Inter-layer Coherence in Double-Layer Quantum-Hall Systems I: Charged Vortices and Kosterlitz-Thouless Phase Transitions
• Symmetry Protected Topological phases of Quantum Matter
• Theta-terms in nonlinear sigma-models
• Theory of Unconventional Spin Density Wave: A Possible Mechanism of the Micromagnetism in U-based Heavy Fermion Compounds
• Topological spin Hall states, charged skyrmions, and superconductivity in two dimensions
• Synthesis, Engineering, and Theory of 2D van der Waals Magnets
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions
• Gauge Invariance, Finite Temperature and Parity Anomaly in D=3
• Direct detection of pure spin-current by x-ray pump-probe measurements
• Induced Parity Breaking Term at Finite Temperature
• Observation of second-harmonic generation induced by pure spin currents
• The Sum Over Topological Sectors and $\theta$ in the 2+1-Dimensional $\mathbb{C}{\mathbb{P}}^1$ $\sigma$-Model
• Colloquium: Quantum Properties and Functionalities of Magnetic Skyrmions
• A proof of the Bloch theorem for lattice models
• Correlation effects and hidden spin-orbit entangled electronic order in parent and electron-doped iridates Sr${}_2$IrO${}_4$
• Chiral non-linear sigma-models as models for topological superconductivity
• Hopf term and the effective Lagrangian for the Skyrmions in a two-dimensional electron gas at small g-factor
• Geometrical phases and quantum numbers of solitons in nonlinear sigma-models
• Odd-parity spin-loop-current order mediated by transverse spin fluctuations in cuprates and related electron systems
• Hopf term induced by fermions
• On the Quantization of the Abelian Chern-Simons Coefficient at Finite Temperature
• Fermionic Hopf solitons and Berry's phase in topological surface superconductors
• The induced Chern-Simons term at finite temperature
• Hopf invariant for long-wavelength skyrmions in quantum Hall systems for integer and fractional fillings
• Topological Aspects of Spin and Statistics in Nonlinear Sigma Models
• Quantum Hall Ferromagnets: Induced Topological term and electromagnetic interactions
• Realization of the topological Hopf term in two-dimensional lattice models
• Spin loop-current textures in Hubbard models