  We perform a mean-field study of the phase diagram of interacting Weyl
semimetals with broken parity, that is, with different densities of right- and
left-handed quasiparticles. As a simple model system, we consider the
Wilson-Dirac Hamiltonian with the chiral chemical potential and on-site
repulsive interactions. We find that the chiral chemical potential somewhat
shrinks the region of the pion condensation (Aoki phase) in the parameter space
of the bare mass and the interaction strength, so that the condensation
thresholds are at smaller interaction strengths. The renormalized chiral
chemical potential monotonously grows with interaction strength everywhere in
the phase diagram, and only the growth rate is discontinuous across the phase
transition lines. These findings are in full agreement with previous results
obtained by one of the authors for the continuum Dirac Hamiltonian, except for
the fact that for our lattice model with explicitly broken chiral symmetry the
boundaries of the Aoki phase remain sharp second-order phase transitions even
at nonzero chiral chemical potential and there are no signatures of Cooper-type
instabilities in the weakly interacting regime.
