The quarkyonic phase and the ZNc symmetry

We investigate the interplay between the ZNc symmetry and the emergence of the quarkyonic phase, adding the flavor-dependent complex chemical potentials μf=μ+iTθf with (θf)=(0,θ,-θ) to the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model. When θ=0, the PNJL model with the μf agrees with the standard PNJL model with the real chemical potential μ. When θ=2π/3, meanwhile, the PNJL model with the μf has the ZNc symmetry exactly for any real μ, so that the quarkyonic phase exists at small T and large μ. Once θ varies from 2π/3, the quarkyonic phase exists only on a line of T=0 and μ larger than the dynamical quark mass, and the region at small T and large μ is dominated by the quarkyonic-like phase in which the Polyakov loop is small but finite.