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Consistent bosonization-debosonization II: The two-lead Kondo problem and the fate of its non-equilibrium Toulouse point

Following the development of a scheme to bosonize and debosonize consistently [N. Shah and C.J. Bolech, Phys. Rev B 93, 085440 (2016); arXiv:1508.03078], we present in detail the Toulouse-point analytic solution of the two-lead Kondo junction model. The existence and location of the solvable point is not modified, but the calculational methodology and the final expressions for observable quantities change markedly as compared to the existent results. This solvable point is one of the remarkably few exact results for non-equilibrium transport in correlated systems. It yields relatively simple analytical expressions for the current in the full range of temperature, magnetic field and voltage. It also shows precisely, within the limitations of the Toulouse fine-tuning, how the transport evolves depending on the relative strengths of inter-lead and intra-lead Kondo exchange couplings ranging from weak to strong. Thus its improved understanding is an important stepping stone for future research.