1. Localized Magnetic States in Metals Miyake Lab. Akiko Shiba Ref.) P. W. Anderson, Phys. Rev. 124 (1961) 41

2. Contents  Introduction Experimental Data  Calculation Hamiltonian Unrestricted Hartree – Fock Approximation Magnetic Case Nonmagnetic Case  Summary

3. Electron Concentration Moment per Fe in Bohr magnetons No localized moment localized moment Experimental Data Magnetic moments of Fe impurity Depend on the host metal Ref.)A.M.Clogston et al., Phys.Rev.125,541(1962) Susceptibility:

4. Hamiltonian free-electron system s-d hybridization repulsive interaction d-states Many-body problem where U E d +U EdEd εFεF V

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

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

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

8. Hartree-Fock Approximation δ↑δ↑ constant is very small, Assume that

9. Hartree-Fock Hamiltonian One-electron Hamiltonian EσEσ

10. DOS of d-electrons Resolvent Green Function ： where DOS of conduction electrons

11. DOS of d-electrons ε EdEd ρｄ（ε）ρｄ（ε） Δ Lorentzian

12. Self-consistent equation Introduce Important parameters! :Self-consistent equation Number of d-electrons:

13. Non-magnetic State (Self-consistency plot) 0.5 Non-magnetic solution Non-magnetic Solution

14. Magnetic State (Self-consistency plot) Magnetic solutions Non-magnetic solution 0.5

15. vs. y=U/Δ magnetic non-magnetic ε εFεF E d +U EdEd (symmetric)

16. vs. y=U/Δ magnetic non-magnetic ε εFεF EdEd E d +U (asymmetric)

17. Magnetic Phase diagram x near 0 x not small or too near 1  Non-magnetic conditions:  Magnetic conditions: x

18. Non-magnetic Case (symmetric) Assume use the approximation: then ε εFεF EdEd E d +U Δ

19. Non-magnetic Case (asymmetric) Valence fluctuation x near 0 Opposite limit: ε εFεF EdEd E d +U Δ

20. Magnetic Case Assume then x not small or too near 1 εFεF E d +U ε EdEd Δ

21. Summary  The Coulomb correlation among d- electrons at the impurity site is important to understand the appearance of magnetic moment.  The existence of magnetic moments depends on ‘ x ’ and ‘ y ’. ε εFεF ε εFεF ε εFεF EdEd E d +U