Log-rise of the Resistivity in the Holographic Kondo Model

We study a single-channel Kondo effect using a recently developed holographic large-$N$ technique. In order to obtain resistivity of this model, we introduce a probe field. The gravity dual of a localized fermionic impurity in 1+1-dimensional host matter is constructed by embedding a localized 2-dimensional Anti-de Sitter (\ads{2})-brane in the bulk of \ads{3}. This helps us construct an impurity charge density which acts as a source to the bulk equation of motion of the probe gauge field. The functional form of the charge density is obtained independently by solving the equations of motion for the fields confined to the \ads{2}-brane. The asymptotic solution of the probe field is dictated by the impurity charge density, which in turn, affects the current-current correlation functions, and hence the resistivity. Our choice of parameters tunes the near-boundary impurity current to be marginal, resulting in a $\log T$ behavior in the UV resistivity, as is expected for the Kondo problem. The resistivity at the IR fixed point turns out to be zero, signaling a complete screening of the impurity.