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Relation between the Anderson model and the s-d model. Ref.) J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149 (1966) 491. Miyake Lab. Akiko Shiba. Outline. Introduction Anderson model s-d model Derivation of the s-d model Perturbation theory Schrieffer- Wolff transformation Summary.
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Relation between the Anderson model and the s-d model Ref.) J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149 (1966) 491 Miyake Lab. Akiko Shiba
Outline • Introduction • Anderson model • s-d model • Derivation of the s-d model • Perturbation theory • Schrieffer- Wolff transformation • Summary
U Ed+U εF V where Ed Anderson Model conduction (s-)electron d-orbital Coulomb repulsion s-d hybridization
ε nd↑=nd↓ No localized moment DOS of conduction electrons Simple Limit: U=0 No Coulomb correlation εF Ed Ed Δ Δ
ε ε Ed+U Ed εF εF Ed+U Ed Simple Limit:Vdk=Vkd=0 No s-d hybridization No localized moment
ε Ed+U εF Coulomb repulsion Ed Simple Limit:Vdk=Vkd=0 No s-d hybridization Ed<εF Ed+U>εF Localized moment appears
s-d Model s:spin operator of the conduction electron S:spin operator of the d-electron Jun Kondo used this model to explain the Kondo effect.
ε Ed+U εF Ed Perturbation theory Anderson Model Existence of localized moment Perturbation theory s-d Model
Perturbation theory Anderson Hamiltonian 2nd order perturbation:
ε Ed+U ε ε εF εF εF Ed Perturbation （ｄ→ｋ→ｄ） 1. 2.
ε Ed+U ε ε εF εF εF Ed Perturbation （ｋ→ｄ→ｋ） 3. 4.
Perturbation Hs-d
ε Ed+U εF Ed Schrieffer- Wolff Transformation Anderson Model Existence of localized moment Schrieffer-Wolff transformation s-d Model
Expand with this S Schrieffer- Wolff Transformation Canonical Transformation If we define S as The first order term in the hybridization will disappear.
Schrieffer- Wolff Transformation Substitute S
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
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 model s-d model equivalent in the local-moment regime
