However, as the SkX is regarded as an interference pattern, it has another degree of freedom, which has been overlooked in the previous researches, the phases of the constituent waves. Similar to an isolated skyrmion, the SkX is characterized by three quantities: skyrmion number, vorticity, and helicity 3. It has attracted enormous attention since the swirling magnetic texture generates an emergent magnetic field through the Berry phase mechanism and results in peculiar transport phenomena, such as the topological Hall effect 3, 9, 10.
Importantly, such a SkX is approximately expressed as a superposition of multiple helical spin density waves, and hence, it can be regarded as an interference pattern by the multiple helices. The magnetic skyrmions often exist in their stable form, the so-called skyrmion crystal (SkX), which is a periodic array of the particle-like magnetic skyrmions. One possible realization was discovered in magnets, in the form of skyrmion-like magnetic textures 4, 5, 6, 7, 8. Although it was originally proposed to explain hadrons in the particle theory 1, 2, it has turned out to be realized in various forms in condensed matter physics 3. The skyrmion is a topological configuration of a continuous field. Our results provide a further diversity of topological spin textures and open a new field of emergent electromagnetism by the phase shift engineering. We demonstrate that such a phase shift can be driven by exchange interactions between the localized spins, thermal fluctuations, and long-range chirality interactions in spin-charge coupled systems. We here theoretically show that a phase shift in the skyrmion crystals leads to a tetra-axial vortex crystal and a meron-antimeron crystal, both of which show a staggered pattern of the scalar spin chirality and give rise to nonreciprocal transport phenomena without the spin-orbit coupling. Although such a phase degree of freedom is relevant to not only magnetism but also transport properties, its effect has not been elucidated thus far.
Since it is regarded as an interference pattern by multiple helical spin density waves, the texture changes with the relative phase shifts among the constituent waves. The magnetic skyrmion crystal is a periodic array of a swirling topological spin texture.
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