Crystalline field effect on the kondo impurity

We consider a Kondo impurity interacting with conduction electrons in two different orbital channels in a crystalline field Δ. The ground state of the impurity is in an orbital singlet, i.e., l _z ^imp = 0, and the valence of the impurity is n ^imp = 2 at half filling without a crystalline field. On the other hand, for an extremely large crystalline field, l _z ^imp - 1/2 and n ^imp = 1. We study the crystalline field effect on the impurity. Neglecting an external magnetic field and a charge fluctuation, we calculate the valence of the impurity analytically via the Bethe-ansatz equations for both weak (Δ < < T _K) and strong (Δ > > T _K) crystalline fields, where T _K is the Kondo temperature. The crystalline field provides the characteristic Kondo logarithms.