Majorana fermions, exact mapping between quantum impurity fixed points with four bulk fermion species, and solution of the “unitarity puzzle”

Abstract

Several quantum impurity problems with four flavors of bulk fermions have zero temperature fixed points that exhibit non-Fermi liquid behavior. These include the two-channel Kondo effect, the two-impurity Kondo model, and the fixed point occurring in the four-flavor Callan-Rubakov effect. We re-interpret the exact conformal field theory (CFT) solution of these fixed points using abelian bosonization, with an extremely simple linear boundary condition on the free bosonic fields. We recover all results of the CFT solution, such as e.g. correlation functions, thermodynamics, partition functions, boundary states, etc. In particular, for the two-channel Kondo fixed point, we derive the single-particle Green function and the square-root singularity of the resistivity using the abelian bosonized formulation. We provide a unified description for all three fixed points, by exploiting the SO(8) symmetry of the four species of bulk fermions. This leads to an exact mapping between correlation functions of the different models. Furthermore, we show that the two-impurity Kondo fixed point and the Callan-Rubakov fixed point are identical theories. All these models have the puzzling property that the S-matrix for scattering of fermions off the impurity seems to be non-unitary. We resolve this paradox by showing that the conduction electrons scatter into (non-local) collective soliton-like excitations, which transform in the spinor representation of the SO(8) Lie algebra; the effect of the quantum impurity can be represented as the ‘triality’ operation of SO(8).