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Duality between zeroes and poles in holographic systems with massless fermions and a dipole coupling

We discuss the zeroes and poles of the determinant of the retarded Green function ($\det G_R$) at zero frequency in a holographic system of charged massless fermions interacting via a dipole coupling. For large negative values of the dipole coupling constant $p$, $\det G_R$ possesses only poles pointing to a Fermi liquid phase. We show that a duality exists relating systems of opposite $p$. This maps poles of $\det G_R$ at large negative $p$ to zeroes of $\det G_R$ at large positive $p$, indicating that the latter corresponds to a Mott insulator phase. This duality suggests that the properties of a Mott insulator can be studied by mapping the system to a Fermi liquid. Finally, for small values of $p$, $\det G_R$ contains both poles and zeroes (pseudo-gap phase).