  We propose a new chirality-imbalance phenomenon arising in baryonic/high
dense matters under a magnetic field. A locally chiral-imbalanced (parity-odd)
domain can be created due to the electromagnetically induced $U(1)_A$ anomaly
in high-dense matters. The proposed local-chiral imbalance generically
possesses a close relationship to a spacial distribution of an inhomogeneous
chiral (pion)-vector current coupled to the magnetic field. To demonstrate such
a nontrivial correlation, we take the skyrmion crystal approach to model
baryonic/high dense matters. Remarkably enough, we find the chirality-imbalance
distribution takes a wave form in a high density region (dobbed
``chiral-imbalance density wave''), when the inhomogeneous chiral condensate
develops to form a chiral density wave. This implies the emergence of a
nontrivial density wave for the explicitly broken $U(1)_A$ current
simultaneously with the chiral density wave for the spontaneously broken
chiral-flavor current. We further find that the topological phase transition in
the skyrmion crystal model (between skyrmion and half-skyrmion phases)
undergoes the deformation of the chiral-imbalance density wave in shape and
periodicity. The emergence of this chiral-imbalance density wave could give a
crucial contribution to studies on the chiral phase transition, as well as the
nuclear matter structure, in compact stars under a magnetic field.

Nuclear Theory;High Energy Physics - Phenomenology