Fig. 3: Interplay between a field-tunable toroidal moment and AHE.

a The Γ₁ spin configuration having a magnetic space group P6₃22, in which the symmetry forbids finite \(\sigma _{{{{\mathrm{xy}}}}}({{{\mathbf{H}}}} = 0)\) and \(M_{{{\mathrm{z}}}}({{{\mathbf{H}}}} = 0)\). b, c The Γ₂ spin configuration having a magnetic space group P6₃22. No symmetry operation forces \(\sigma _{{{{\mathrm{xy}}}}}\) and \(M_{{{\mathrm{z}}}}\) to vanish. Three sky-blue ellipses in a, b denote the three-fold easy-axes to realize the Γ₁ or Γ₂ ground state, respectively. d Total energy from the DFT calculation as a function of the uniform spin rotation angle \(\theta\) shown in a. e Comparison between a toroidal moment along the c-axis (a solid blue line) and calculated \(\sigma _{xy}\) for the spin configurations specified by \(\theta\) in a (black circles). f Interpreting the sign change of AHE as a switch of the toroidal moment. A blue (orange) rectangle denotes a region of the \(\Gamma _2^ +\) (\(\Gamma _2^ -\)) spin configuration.