Skyrmions in a magnetic field and ${\pi }^0$ domain wall formation in dense nuclear matter

We elucidate magnetic effects in the skyrmion system to probe into the dense nuclear matter. We find a deformed ${\pi }^0$ dipole structure of a baryon induced by a magnetic field. We then extend our scope to stacked Skyrme crystal layers to scrutinize phases of magnetized nuclear matter. We observe a quantized magnetic flux and identify a phase transition from a crystalline state to a ${\pi }^0$ domain wall corresponding to a topological transmutation from ${\pi }_3(S^3)$ to ${\pi }_1(S^1)$. We establish the phase diagram, which could be explored also in analogous systems with two-component Bose-Einstein condensates.