IEP Kosice, 16 May 2007

1. IEP Kosice, 16 May 2007 Two-band Hubbard model of superconductivity: physical motivation and Green function approach to the solution Gh. Adam, S. Adam LIT-JINR Dubna and IFIN-HH Bucharest Gh. Adam and S. Adam, Rigorous derivation of the mean field Green functions of thetwo-band Hubbard model of superconductivity, arXiv:0704.0692v1 [cond-mat.supr-con] Subm. to J.Phys. A: Math. Gen

2. OUTLINE I. Physical Motivation II. Model Hamiltonian III. Mean Field Approximation IV. Reduction of Correlation Order of Processes Involving Singlets V. Frequency Matrix and Green Function in Reciprocal Space VI. DISCUSSION

3. I. Physical Motivation

4. Damascelli et al., RMP, 75, 473, 2003

5. Left: Elementary cell. Right: 3D Brillouin zone (body-centered tetragonal) and its 2D projections. Diamond: Fermi surface at half filling calculated with only the nearest neighbor hopping; Gray area: Fermi surface obtained including also the next-nearest neighbor hopping. Note that is the midpoint along Γ−Ζ is not a true symmetry point. Crystal structure and Fermi surface of La2-xSrxCuO4 (LSCO) (after Damascelli et al., RMP, 75, 473, 2003)

6. i j Effective Spin States • Schematic representation of the cell distribution within CuO2 plane • Antiferromagnetic arrangement of the spins of the holes at Cu sites • Effect of the disappearance of a spin within spin distribution

7. xz, yz Crystal field splitting and hybridization giving rise to the Cu-O bands (Fink et al., IBM J. Res. Dev., 33, 372, 1989).

8. Qualitative illustration of the electronic density of states of the p-d model with three bands: bonding (B), anti-bonding (AB), and non-bonding (NB). (c) metallic state at half-filling of AB band for U = 0 (see (a) on previous slide) (d) Mott-Hubbard insulator for Δ > U> W [W ~ 2eV is the width of AB band] (e) charge-transfer insulator for U > Δ > W (f) charge-transfer insulator for U > Δ > W,with the two-hole p-d band split into the triplet (T, S=1) and the Zhang-Rice singlet (ZRS, S=0) bands. (after Damascelli et al., RMP, 75, 473, 2003)

9. Peculiarity of the hole-singlet band structure If (a spin state at site i belongs to the hole subband ) then it is theuniquely occupied state at site i [|i  in hole subband excludes the presence of |i ; |i in hole subband excludes the presence of |i ] If (a spin state at site i belongs to the singlet subband ) then theopposite spin state is also presentat site i . State description in terms of Hubbard operators is able to handle consistently these requirements.

10. Basic Results of Analysis t-J Model Effective parameters fora single subband(whichintersects the Fermi level). • Describessuperconducting state • Unable to describenormal state ═> Misses consistent description of phase transition Effective parameters fortwo subbands(whichlay nearest to Fermi level). Hubbard operator algebra preserves the Pauli exclusion principle May describe both superconducting and normal states ═> Consistent description of phase transition Over- simplification Two-band Hubbard Simplest consistent model

11. Previous Results of Hubbard Model Studies

12. Two-subband effective Hubbard model:AFM exchange pairing W e2 m 0 t12 e1 j i Estimate in WCA gives for Tcex : N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)

13. W ws m -ws 0 Two-subband effective Hubbard model:Spin-fluctuation pairing e2 e1 i j Estimate in WCA gives for Tcsf: N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)

14. Critical Temperature Tc(δ)(teffunits) Total Contribution to Tc Exchange Contribution Kinematic Interaction(spin fluctuation) N.M. Plakida et al. JETP, 97, 331 (2003)

15. II. Model Hamiltonian

16. The Hamiltonian N.M.Plakida, R.Hayn, J.-L.Richard, PRB, 51, 16599, (1995) N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)

17. Properties of Hubbard Operators

18. Hubbard Operators (1)

19. Hubbard Operators (2)

20. Hubbard Operators (3)

21. End [Properties of Hubbard Operators]

22. Hubbard p-Form of labels

23. Hubbard 1- Forms in Hamiltonian

24. The Hamiltonian in terms of Hubbard 1-forms

25. Energy Parameters (1)

26. Energy Parameters (2)

27. Energy Parameters (3)

28. Hopping contributions to the Hamiltonianin terms of Hubbard linear forms

29. III. Mean Field Approximation

33. Consequences of translation invariance of the spin lattice

35. Mean Field Approximation

36. Frequency matrix under spin reversal

37. Deriving spin reversal invariance properties

40. Normal one-site statistical averages

41. Anomalous one-site statistical averages

42. Two-site statistical averages

43. Need oftwo kindsof particle number operators At a given lattice site i, there is a single spin state of predefined spin projection. The total number of spin states equals 2. The conventional particle number operator Ni provides unique characterization of the occupied states within the model.

44. Frequency matrix in (r,ω)-representation

45. Frequency Matrix in (r,ω)-representation

46. The Normal Hopping Matrix

47. Consequences of spin reversal invariance

48. The Anomalous Hopping Matrix

49. IV. Reduction of Correlation Order of Processes Involving Singlets

50. Energy parameters (hole-doped cuprates)