Holographic fermions with running chemical potential and dipole coupling

We explore the properties of the holographic fermions in extremal $R$-charged black hole background with a running chemical potential, as well as the dipole coupling between fermions and the gauge field in the bulk. We find that although the running chemical potential effect the location of the Fermi surface, it does not change the type of fermions. We also study the onset of the Fermi gap and the gap effected by running chemical potential and the dipole coupling. The spectral function in the limit $\omega\rightarrow0$ and the existence of the Fermi liquid are also investigated. The running chemical potential and the dipole coupling altogether can make a non-Fermi liquid become the Landau-Fermi type.