Fig. 1: Effective time-reversal symmetry.
From: Magnetic geometry induced quantum geometry and nonlinear transports
a–c Effective time-reversal symmetries always emerge in collinear (a) and coplanar AFMs (b), with \({U}_{\perp }\left(\pi \right)T\) and \({U}_{\perp }^{{\prime} }\left(\pi \right)T\), respectively, and may emerge in certain noncoplanar AFMs (c) with \(T{{\boldsymbol{\tau }}}\). d–f Combined symmetry of spatial inversion and time-reversal \({PT}\) is available in certain collinear AFMs (d), while effective combined symmetry can emerge in coplanar (e), and noncoplanar AFMs (f) with \({U}_{{{\bf{n}}}}\left(\pi \right){TP}\). \(T\): time-reversal; \(P\): spatial inversion; \({U}_{\perp }\left(\pi \right)\): two-fold spin rotation along an axis normal to the Néel vector; \({U}_{\perp }^{{\prime} }\left(\pi \right)\): two-fold spin rotation along the axis normal to all in-plane magnetic moments; \({U}_{{{\bf{n}}}}\left(\pi \right)\): two-fold spin rotation along \({{\bf{n}}}\) axis; \({{\boldsymbol{\tau }}}\): fractional lattice translation. Red arrows and yellow balls denote magnetic moments and atoms, respectively, and shadowed arrows represent the intermediate state of magnetic moments operated by part of the combined symmetry, i.e., \(T\) in (a–c) and \(P\) in (d–f).
