Local susceptibility and Kondo scaling

The Kondo scale $T_K$ for quantum impurity systems is typically assumed to guarantee universal scaling of physical quantities. In practice, however, not every definition of $T_K$ necessarily supports this notion away from the strict scaling limit for finite bandwidth $D$. Various theoretical definitions of $T_K$ are analyzed based on the inverse magnetic impurity susceptibility at zero temperature. While conventional definitions in that respect quickly fail to ensure universal Kondo scaling for all $D$, an altered definition of $T_K^{\mathrm{sc}}$ is presented which allows universal scaling of dynamical or thermal quantities for a given fixed Hamiltonian. If the scaling is performed with respect to an external parameter which directly enters the Hamiltonian, such as magnetic field, the corresponding $T_K^{\mathrm{sc,B}}$ for universal scaling may differ, yet becomes equivalent to $T_K^{\mathrm{sc}}$ in the scaling limit. The only requirement for universal scaling in the full Kondo parameter regime with a residual error of less than $1\%$ is a well-defined isolated Kondo feature with $T_K\leq 0.01\,D$. By varying $D$ over a wide range relative to the bare energies of the impurity, this allows a smooth transition from the Anderson to the Kondo model.