Two-point Functions in a Holographic Kondo Model

We develop the formalism of holographic renormalization to compute two-point functions in a holographic Kondo model. The model describes a $(0+1)$-dimensional impurity spin of a gauged $SU(N)$ interacting with a $(1+1)$-dimensional, large-$N$, strongly-coupled Conformal Field Theory (CFT). We describe the impurity using Abrikosov pseudo-fermions, and define an $SU(N)$-invariant scalar operator $\mathcal{O}$ built from a pseudo-fermion and a CFT fermion. At large $N$ the Kondo interaction is of the form $\mathcal{O}^{\dagger} \mathcal{O}$, which is marginally relevant, and generates a Renormalization Group (RG) flow at the impurity. A second-order mean-field phase transition occurs in which $\mathcal{O}$ condenses below a critical temperature, leading to the Kondo effect, including screening of the impurity. Via holography, the phase transition is dual to holographic superconductivity in $(1+1)$-dimensional Anti-de Sitter space. At all temperatures, spectral functions of $\mathcal{O}$ exhibit a Fano resonance, characteristic of a continuum of states interacting with an isolated resonance. In contrast to Fano resonances observed for example in quantum dots, our continuum and resonance arise from a $(0+1)$-dimensional UV fixed point and RG flow, respectively. In the low-temperature phase, the resonance comes from a pole in the Green's function of the form $-i \langle {\cal O} \rangle^2$, which is characteristic of a Kondo resonance.