Chiral spiral in the presence of chiral imbalance

The phase diagram of the two-dimensional Nambu–Jona-Lasinio (NJL2) (or chiral Gross-Neveu) model is characterized by an order parameter in the form of a chiral spiral. Its radius vanishes at a critical temperature, and its period depends only on the chemical potential. We generalize these findings to chirally imbalanced systems by including a chiral chemical potential μ5. The relationship between the present static approach and a previous, time-dependent one is traced back to a half-local symmetry which the NJL2 model shares with massless Dirac fermions, but which has been neglected so far. The structure of chiral spiral matter is further elucidated by computing fermion and antifermion momentum distribution functions, using a Bogoliubov transformation.