1. LIT-JINR Dubna and IFIN-HH Bucharest
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 the two-band Hubbard model of superconductivity, arXiv: v1 [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. Crystal structure and Fermi surface of La2-xSrxCuO4 (LSCO)
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. Effective Spin States i j
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. Crystal field splitting and hybridization giving rise to the
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. (after Damascelli et al., RMP, 75, 473, 2003)
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 the uniquely 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 the opposite spin state is also present at 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 for a single subband (which intersects the Fermi level).
Describes superconducting state
Unable to describe normal state
═> Misses consistent description of phase transition
Effective parameters for two subbands (which lay 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 pairinge1 i j
Estimate in WCA gives for Tcex :
N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)

13. Two-subband effective Hubbard model: Spin-fluctuation pairing
-ws ws W m 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) (teff units) Total Contribution to Tc
Exchange Contribution
Kinematic Interaction (spin fluctuation)
N.M. Plakida et al. JETP, 97, 331 (2003)

15. Dispersion Curves (a) and Spectral Functions (b)
δ=0.05
δ=0.3
N.M.Plakida, V.S. Oudovenko JETP, 104, 230 (2007)

16. II. Model Hamiltonian

17. 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)

18. Properties of
Hubbard Operators

19. Hubbard Operators (1)

20. Hubbard Operators (2)

21. Hubbard Operators (3)

22. [Properties of Hubbard Operators]
End
[Properties of Hubbard Operators]

23. Hubbard p-Form of labels

24. Hubbard 1- Forms in Hamiltonian

25. The Hamiltonian in terms of Hubbard 1-forms

26. Energy Parameters (1)

27. Energy Parameters (2)

28. Energy Parameters (3)

29. Hopping contributions to the Hamiltonian in terms of Hubbard linear forms

30. III. Mean Field Approximation

34. Consequences of translation invariance of the spin lattice

36. Mean Field Approximation

37. Frequency matrix under spin reversal

38. Deriving spin reversal invariance properties

42. Normal one-site statistical averages

43. Anomalous one-site statistical averages

44. Two-site statistical averages

45. Need of two kinds of 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.

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

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

48. The Normal Hopping Matrix

49. Consequences of spin reversal invariance

50. The Anomalous Hopping Matrix

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

52. Energy parameters (hole-doped cuprates)

53. For hole-doped cuprates, the Spectral Theorem gives:

54. Result of reduction of correlation order

55. Energy parameters (electron-doped systems)

56. For electron-doped cuprates, we use the second form of the Spectral Theorem to get exponentially small terms:

57. GMFA Correlation Functions for Singlet Hopping

58. GMFA Correlation Functions for Superconducting Pairing

59. V. Frequency Matrix and Green Function in Reciprocal Space

60. Frequency Matrix in (q, ω)-representation (1)

61. Frequency Matrix in (q, ω)-representation (2)

62. Frequency Matrix in (q, ω)-representation (3)

63. Frequency Matrix in (q, ω)-representation (4)

64. Frequency Matrix in (q, ω)-representation (5)

65. GMFA Green function matrix in (q, ω)-representation (1)

66. GMFA-GF Matrix in (q, ω)-representation (2)

67. GMFA Energy Spectrum

68. VI. DISCUSSION

69. 1.1 We considered the effective two-band Hubbard model of high-Tc superconductivity in cuprates [N.M. Plakida et al. PRB 51, (1995); ZhETF 124, 367 (2003)/JETP 97, 331 (2003)]
1.2 We studied consequences following from the algebra of the Hubbard operators.
1.3 We derived rigorous consequences following from:
- spin lattice translation invariance
- invariance under spin reversal
1.4 The order of boson-boson correlation functions describing superconducting pairing and singlet hopping within Mean Field Approximation of the Green function solution of the model was reduced
1.5 Next step is the study of effects following from the variation of the parameters of the model

70. Thank you for your attention !