1. Interplay between spin, charge, lattice and orbital degrees of freedom
Lecture notes Les Houches June 2006
George Sawatzky

2. LSDA+U
Simplified version :
V.I. Anisimov et al., PRB 44, 943 (1991)

3. Czyzk et l PRB 49,
14211(1994)
LSDA+U antiferromagnetic S=.8 Bohr magnetons, E gap = 1.65 eV
LSDA+U
LSDA
LSDA+U also has no electron correlation
Single Slater det. of Bloch states. No multiplets.

4. Problems with LDA+U for metallic systems
Ferromagnetic
Note U gap closes with doping
No spectral weight transfer

6. Exact diagonalization in 1D Hubbard 10 sites U=10t
U Gap increases with doping
Spectral weight is transfered from
upper H band to the lower H band
Meinders et al, PRB 48, 3916 (1993)

7. Mott – Hubbard U N N PES EF PES N-1 N+1 2 2 EF EF Spectral weight
transfer U N N
Remove one electron
Create two addition states
At low energy
PES EF
PES
N-1
N+1 2 2 EF EF

8. These particles block 2 or more states
Bosons – block 0 states
Fermions – block 1 state
Integral of the low
Energy spectral weight
For electron addition if
Hole doped (left) and
Electron removal for e
Doped (right side)

9. Eskes et al PRL 67, (1991) 1035
Meinders et al PRB 48, (1993) 3916

10. Exact diagonalization 1D Hubbard
Meinders et al,
PRB 48, 3916 (1993)

11. Don’t know of a rigorous
Proof of Hubb---t,J (U>>w)

12. Spin charge separation in 1D
Antiphase
Domain wall
Now the charge is free to move

13. Magnons and spinons in 1D
Two spinons
Spinons propagate via J Si+S-1+1

15. Similar is some sense to the 1D case it is proposed that one has
2D rivers of charge separating anti-phase domain walls.
Charges can now fluctuate from left to right without costing J
Anisimov, Zaanen ,Andersen, Kivelson,Emery-----

16. Closer to real systems

17. Oxides
Remember at surfaces U is increased, Madelung is decreased, W is decreased

18. For divalent cations

19. This seems to happen in CrO2 where the O bands cross the
If the charge transfer gap becomes negative we will get a strange metal
This seems to happen in CrO2 where the O bands cross the
Fermi driving the system to a half metallic ferromagnet
Korotin PRL 80 (1998) 4305

20. 3 most frequently used methods
Anderson like impurity in a semiconducting host consisting of full O 2p bands and empty TM 4s bands including all multiplets
Developed for oxides in early 1980’s, Zaanen, Sawtzky, Kotani, Gunnarson,-----
Cluster exact diagonalization methods. O cluster of the correct symmetry with TM in the center. Again include all multiplets crystal fields etc
Developed for oxides in early 1980’s Fujimori, Sawatzky,Eskes,
Dynamic Mean Field methods, CDMFT, DCA which to date do not include multiplets
Developed in the late 1990’s: Kotliar, George, Vollhart---

21. Anderson impurity ansatz Like DMFT but not self consistant
Zaanen et al prl
(1985)
Anderson impurity ansatz Like DMFT but not self consistant
But also including all multiplet interactions
Kondo resonance

22. To calculate the gap we calculate the ground state of the system with n,n-1, and n+1 d electrons Then the gap is E(Gap)= E(n-1)+E(n+1)-2E(n)

23. Two new complications
d(n) multiplets determined by Slater atomic integrals or Racah parameters A,B,C for d electrons. These determine Hund’s rules and magnetic moments
d-o(2p) hybridization ( d-p hoping int.) and the o(p)-o(p) hoping ( o 2p band width) determine crystal field splitting, superexchange , super transferred hyperfine fields etc.

24. More general multiband model Hamiltonian

25. We usually take U(pp) =0 although it is about 5 eV as
Measured with Auger but the O 2p band is usuallu fiull or
nearly full.

26. Ways to “screen “ or rather reduce Uor F0
U in Cu atom is 18eV in the solid 8eV
In a polarizable medium
“Solvation” in chemistry

28. Rest comes from bond
Polarization involving
O 2p and TM 4s states

29. As we remove or add d electrons charge moves from
O(2p) to or from TM(4s) reducing the d electron
Removal energy as well as the d electron addition energy.
Reduces U effectively by about 6-8 eV.
Recall though that these effects will yield satellites or incoherent
Parts to the spectral function at energies corresponding to the
O(2p)-TM(4s) energy splitting.

30. Note that B and C are only slightly reduced in the solid they do not
involve changes in the local charge !!!

31. For the N-1 electron states we need d8, d9L, d10L2 where L
denotes a hole in O 2p band. The d8 states exhibit multiplets

32. Anderson Impurity calculation
H. Eskes and G.A. Sawatzky
PRL 61, 1415 (1988).
Anderson Impurity calculation
Zhang Rice
singlet

33. Photoemission spectrum of CuO
J. Ghijsen et al
Phys. Rev. B. 42, (1990) 2268.
Photoemission spectrum of CuO
Energy below Ef in eV

35. Example of two cluster calculations to obtain the parameters
For a low energy theory ( single band Hubbard or tJ )

36. Antibonding splitting Measure d-d hoping
Eskes etal PRB
44,9656, (1991)
0 to 1 hole spectrum
One of the Cu’s is d9
The other d10 in the
Final state. Bonding
Antibonding splitting
Measure d-d hoping
1 hole to 2 holes final
State is od9 on both
Cu’s Triplet singlet
Splitting yields super
Exchange J
2 holes to 3 holes final
state is d9 for both Cu’s
Plus a hole on O forming
A singlet (ZR) with one of
The Cu’s . Splitting in red
Yields the ZR-ZR hoping
integral as in tJ

37. Need multiband models to describe TM compounds
However numerous studies have shown that this can sometimes be reduced to an effective single band Hubbard model at least for highTc’s BUT ONLY FOR LOW ENERGY EXCITATIONS E<0.5eV
Macridin et al Phys. Rev. B 71, (2005)

39. (Maximize spin)

43. Crystal and ligand field splittings
Often about 0.5 eV
In Oh symmetry

44. Eg-O2p hoping is 2 times as large as T2g-O-2p hoping
Often about 1-2eV
In Oxides

46. High Spin – Low Spin transition very common in
Co(3+)(d6), as in LaCoO3, not so common in Fe(2+)(d6)
Because of the smaller hybridization with O(2p)

47. Mixed valent system could lead to strange effects
Such as spin blockade for charge transport and high
thermoelectric powers

48. What would happen if 2Jh <10Dq<3Jh
If we remove one electron from d6 we would go from
S=0 in d6 to S=5/2 in d5. The “hole “ would carry a spin
Of 5/2 as it moves in the d6 lattice.

51. If the charge transfer energy gets small we have to
Modify the superexchange theory
Anderson 1961
New term