  The skyrmion crystal is a periodic array of a swirling topological spin
texture. Since it is regarded as an interference pattern by multiple helical
spin density waves, the texture changes with the relative phases among the
constituent waves. Although the phase degree of freedom is relevant to not only
magnetism but also transport properties, its effect has not been elucidated
thus far. We here theoretically show that a phase shift can occur in the
skyrmion crystals to stabilize tetra-axial vortex crystals with staggered
scalar spin chirality. This leads to spontaneous breaking of the lattice
symmetry, which results in nonreciprocal transport phenomena even without the
relativistic spin-orbit coupling. We show that such a phase shift is driven by
long-range chirality interactions or thermal fluctuations in spin-charge
coupled systems.
