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 Skyrmion and quantify nontrivial rearrangements of the confining pressure distribution. We confirm an isospin-dependent baryon spectrum from the anomaly-induced action. 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.