On the Role of the Canonical Transformation in the Single-Channel Kondo Model

Abstract

This pedagogical work presents the significant role that canonical transformation plays in the interpretation of the Abelian bosonized single-channel SU(2) Kondo model, emphasizing its effect on the scaling dimension $\Delta$. The transformation shifts the longitudinal exchange coupling and modifies the scaling dimension of the spin-flip vertex $\tau_{\pm} e^{\pm i\beta \phi}$. Rather than fixing $\Delta$ to the fermionic value $\tfrac{1}{2}$, we keep $\alpha$ explicit, which allows us to identify how different choices lead to marginal or relevant regimes through $(1-\Delta(\alpha))J_\perp$. This approach offers a direct way to trace the scaling behavior from the bosonized Hamiltonian and shows how the RG flow connects to the definition of the Kondo temperature, where the resistance diverges, without switching to other methods.