Antisymmetric breathing in altermagnetic skyrmions

A skyrmion in an altermagnet with \(d\) wave symmetry consists of two elliptical sublattice textures with perpendicular long axes. For each sublattice component $η=A,B$, we define an effective skyrmion radius $R_η=\sqrt{a_ηb_η}$, where $a_η$ and $b_η$ are the distances from the skyrmion center to it