Particle-hole instability in the $AdS_4$ holography

We show that particle-hole pairing is realized in the background of a charged black hole in magnetic field. The pairing instability occurs for sufficiently large fermion charges, which correspond to the Fermi liquid regime. The critical temperature for Fermi liquids is proportinal to the magnetic field and vanishes as we approach the non-Fermi liquid state. The pairing order parameter leads to a relative shift of the Fermi surfaces corresponding to the bulk fermions with spin up and down. The value of the shift in Fermi momentum $k_F$ and the critical temperature $T_c$ are proportional to the effective density of states at the Fermi surface. Our one-loop calculations provide a dual description of the magnetic catalysis for the lowest Landau level in graphene. This analyses may be relevant for the antiferromagnetic behavior in the cuprate superconductors and for the chiral spirals in the chiral magnetic effect. We also discuss thermodynamic and transport properties of a system at the boundary at zero magnetic field. The scaling behavior of the specific heat is $c\sim T$ for Fermi liquid and $c\sim T^{2\nu}$ for non-Fermi liquid, while the behavior of the DC conductivity is the same $\sigma\sim T^{-2\nu}$ in both cases. While it can be difficult to extract transport and hydrodynamic from the lattice, the $AdS/CFT$ approach provides a robust frame for nonperturbative calculation of these properties.