1. Relation between the Anderson model and the s-d model Miyake Lab. Akiko Shiba Ref.) J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149 (1966) 491

2. Outline  Introduction Anderson model s-d model  Derivation of the s-d model Perturbation theory Schrieffer- Wolff transformation  Summary

3. Anderson Model conduction (s-)electron s-d hybridization Coulomb repulsion d-orbital where U E d +U EdEd εFεF V

4. Simple Limit: U=0 No Coulomb correlation No localized moment n d ↑ =n d ↓ ε εFεF EdEd EdEd Δ Δ DOS of conduction electrons

5. Simple Limit:V dk =V kd =0 No s-d hybridization ε εFεF EdEd E d +U ε εFεF EdEd No localized moment

6. Simple Limit:V dk =V kd =0 No s-d hybridization Localized moment appears ε εFεF EdEd E d +U Coulomb repulsion E d <ε F E d +U>ε F

7. s-d Model Jun Kondo used this model to explain the Kondo effect. s:spin operator of the conduction electron S:spin operator of the d-electron

8. Perturbation theory Anderson Model s-d Model Perturbation theory Existence of localized moment ε εFεF EdEd E d +U

9. Perturbation theory Anderson Hamiltonian 2nd order perturbation:

10. Perturbation （ｄ → ｋ → ｄ） ε εFεF ε εFεF EdEd E d +Uε εFεF 1. 2.

11. Perturbation （ｋ → ｄ → ｋ） 3. ε εFεF ε εFεF EdEd E d +Uε εFεF 4.

12. Perturbation H s-d

13. Schrieffer- Wolff Transformation Anderson Model s-d Model Schrieffer-Wolff transformation Existence of localized moment ε εFεF EdEd E d +U

14. Schrieffer- Wolff Transformation Canonical Transformation If we define S as The first order term in the hybridization will disappear. Expand with this S

15. Schrieffer- Wolff Transformation

16. Substitute S

17. Schrieffer- Wolff Transformation s-d exchange interaction potential scattering energy shift of d-orbital and renormalization of Coulomb repulsion changes the occupancy of the d-orbital by 2electrons

18. Summary  In the limit of small s-d mixing, which is the most favorable case for the occurrence of a moment, the Anderson model and the s-d model are equivalent.  The s-d model was able to be derived from the Anderson model which is a model more fundamental. Anderson models-d model equivalent in the local-moment regime

19. Schrieffer- Wolff Transformation

20. (σ ： Pauli matrices)