Fig. 2: The emergence of chiral and crystal Hall effect of ferro- and antiferromagnets on a honeycomb lattice.

a The definition of the angles used to characterize the canted spin structure of spins s_A and s_B. The initial direction of collinear magnetization \({\hat{{\bf{s}}}}_{0}=({\theta }_{0},{\varphi }_{0})\) with polar angle θ₀ and azimuthal angle φ₀ is kept constant during canting, \({\hat{{\bf{s}}}}_{0} \sim {{\bf{s}}}_{{\rm{A}}}+{{\bf{s}}}_{{\rm{B}}}\). The spins are canted in the plane of constant φ₀ by an angle θ for s_A and −θ for s_B with respect to \({\hat{{\bf{s}}}}_{0}\). The changes in the bandstructure of the ferromagnetic (FM) (b) and antiferromagnetic (AFM) (c) spins initially along \({\hat{{\bf{s}}}}_{0}=(10{0}^{\circ },1{0}^{\circ })\) upon canting by ±10^∘. The thin gray line with circles marks the initial bandstucture while blue and red lines mark the bandstructure for θ = 10^∘ and θ = −10^∘, respectively. The corresponding anomalous Hall conductivity (AHC), σ_xy, as a function of the Fermi energy is shown for the FM (d) and AFM (e) cases for positive (solid blue line) and negative (dashed red line) canting. The symmetric, \({\sigma }_{xy}^{s}\), and antisymmetric, \({\sigma }_{xy}^{a}\), parts of the AHC are shown with dark orange and dark blue lines. All values are in e²/h, where e is the elementary charge and h is Planck’s constant. f–k While for the high-symmetry direction of \({\hat{{\bf{s}}}}_{0}=(10{0}^{\circ },{0}^{\circ })\) the symmetry properties of the Berry curvature of the first two bands in the FM case, Ω^a(10^∘, k), lead to vanishing overall chiral Hall effect (f), the breaking of symmetry for \({\hat{{\bf{s}}}}_{0}=(10{0}^{\circ },1{0}^{\circ })\) results in a net effect (g). The complex structure of Ω^a(10^∘, k) of the first band from (c) in k-space, (h), is clearly correlated with the separation between the first and second bands in energy, shown in k.