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

TL;DR

The work analyzes the Abelian bosonization of the single-channel Kondo model, demonstrating that a canonical transformation shifts the longitudinal exchange and modifies the scaling dimension of the spin-flip vertex. By keeping the transformation parameter $α$ explicit, it derives $Δ(α) = \frac{(\sqrt{2}-α)^2}{2}$ and shows how $(1-Δ(α))J_⊥$ governs RG flow toward or away from strong coupling, tying this to the Kondo temperature scale. The approach clarifies the link between the bosonized Hamiltonian and RG dynamics, including the Toulouse point as a refermionized limit, and provides a direct route to understanding the temperature dependence of the resistance via $T_K$ without changing methods. Overall, the paper offers a transparent, step-by-step derivation of how canonical transformations affect scaling and thermodynamics in the Kondo problem, with explicit α-dependence enabling exploration of diverse physical regimes. It emphasizes that the canonical transformation is not merely a mathematical tool but a lens for tracing scaling behavior across parameter space.

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 $Δ$. The transformation shifts the longitudinal exchange coupling and modifies the scaling dimension of the spin-flip vertex $τ_{\pm} e^{\pm iβφ}$. Rather than fixing $Δ$ to the fermionic value $\tfrac{1}{2}$, we keep $α$ explicit, which allows us to identify how different choices lead to marginal or relevant regimes through $(1-Δ(α))J_\perp$. 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.