Spectral function of a localized fermion coupled to the Wilson-Fisher conformal field theory

We describe the dynamics of a single fermion in a dispersionless band coupled to the 2+1 dimensional conformal field theory (CFT) describing the quantum phase transition of a bosonic order parameter with N components. The fermionic spectral functions are expected to apply to the vicinity of quantum critical points in two-dimensional metals over an intermediate temperature regime where the Landau damping of the order parameter can be neglected. Some of our results are obtained by a mapping to an auxiliary problem of a CFT containing a defect line with an external field which locally breaks the global O$(N)$ symmetry.