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IEP Kosice, 16 May 2007IEP Kosice, 16 May 2007
Two-band Hubbard model Two-band Hubbard model of superconductivity: of superconductivity: physical motivation and physical motivation and Green function approach Green function approach to the solutionto the solution
Gh. Adam, S. AdamLIT-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:0704.0692v1 [cond-mat.supr-con] Subm. to J.Phys. A: Math. Gen
OUTLINEOUTLINEI. 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
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
I. Physical MotivationI. Physical Motivation
Damascelli et al., RMP, 75, 473, 2003
Crystal structure and Fermi surface of La2-xSrxCuO4 (LSCO)(after Damascelli et al., RMP, 75, 473, 2003)
Left: Elementary cell.Right: 3D Brillouinzone (body-centeredtetragonal) andits 2D projections.Diamond: Fermisurface at half fillingcalculated with onlythe nearest neighborhopping; Gray area: Fermisurface obtainedincluding also thenext-nearest neighborhopping.Note that is themidpoint along Γ−Ζis not a truesymmetry point.
(a) Schematic representation of the cell distribution within CuO2 plane
(b) Antiferromagnetic arrangement of the spins of the holes at Cu sites(c) Effect of the disappearance of a spin within spin distribution
Effective Spin StatesEffective Spin States
i
j
Crystal field splitting and hybridization giving rise to theCu-O bands (Fink et al., IBM J. Res. Dev., 33, 372, 1989).
xz, yz
Qualitative illustration of the electronic density of states of the p-d model withthree 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)
Peculiarity of the hole-singletband structure
Peculiarity of the hole-singletband 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 operatorsis able to handle consistently these requirements.
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 operatorsis able to handle consistently these requirements.
Basic Results of AnalysisBasic Results of Analysis
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
t-J Model
Two-bandHubbard
Over-simplification
Simplestconsistent
model
Previous Results of Hubbard
Model Studies
Previous Results of Hubbard
Model Studies
i
t12
2
1
0
W
Two-subband effective Hubbard model: AFM exchange pairing
Two-subband effective Hubbard model: AFM exchange pairing
Estimate in WCA gives for Tcex :
j
N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)
2
i j
10
s
s
W
Estimate in WCA gives for Tcsf :
Two-subband effective Hubbard model: Spin-fluctuation pairingTwo-subband effective Hubbard model: Spin-fluctuation pairing
N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)
Critical Temperature
Critical Temperature
Tc(δ) (teff units)Tc(δ) (teff units)
Kinematic Interaction (spin fluctuation)Kinematic Interaction (spin fluctuation)
Exchange ContributionExchange Contribution
Total Contribution to Tc Total Contribution to Tc
N.M. Plakida et al. JETP, 97, 331 (2003)
II. Model Hamiltoni
an
II. Model Hamiltoni
an
The HamiltonianThe 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)
Properties of
Hubbard Operators
Properties of
Hubbard Operators
Hubbard Operators (1)Hubbard Operators (1)
Hubbard Operators (2)Hubbard Operators (2)
Hubbard Operators (3)Hubbard Operators (3)
End[Properties of Hubbard Operators]
End[Properties of Hubbard Operators]
Hubbard p-Form of labels Hubbard p-Form of labels
Hubbard 1- Forms in Hamiltonian Hubbard 1- Forms in Hamiltonian
The Hamiltonian in terms of Hubbard 1-forms
The Hamiltonian in terms of Hubbard 1-forms
Energy Parameters (1)Energy Parameters (1)
Energy Parameters (2)Energy Parameters (2)
Energy Parameters (3)Energy Parameters (3)
Hopping contributions to the Hamiltonianin terms of Hubbard linear forms
Hopping contributions to the Hamiltonianin terms of Hubbard linear forms
III. Mean Field ApproximationIII. Mean Field Approximation
Consequences of translation invariance of the spin lattice
Consequences of translation invariance of the spin lattice
Mean Field ApproximationMean Field Approximation
Frequency matrix under spin reversalFrequency matrix under spin reversal
Deriving spin reversal invariance propertiesDeriving spin reversal invariance properties
Normal one-site statistical averagesNormal one-site statistical averages
Anomalous one-site statistical averagesAnomalous one-site statistical averages
Two-site statistical averagesTwo-site statistical averages
Need of two kinds of particle number operators
Need of two kinds of particle number operators
At a given lattice site i, there is a single spin state of predefinedspin 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.
At a given lattice site i, there is a single spin state of predefinedspin 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.
Frequency matrix in
(r,ω)-representati
on
Frequency matrix in
(r,ω)-representati
on
Frequency Matrix in (r,ω)-representation
Frequency Matrix in (r,ω)-representation
The Normal Hopping MatrixThe Normal Hopping Matrix
Consequences of spin reversal invarianceConsequences of spin reversal invariance
The Anomalous Hopping MatrixThe Anomalous Hopping Matrix
IV. Reduction of Correlation
Order of Processes Involving Singlets
IV. Reduction of Correlation
Order of Processes Involving Singlets
Energy parameters (hole-doped cuprates)Energy parameters (hole-doped cuprates)
For hole-doped cuprates, the Spectral Theorem gives:For hole-doped cuprates,
the Spectral Theorem gives:
Result of reduction of correlation orderResult of reduction of correlation order
Energy parameters (electron-doped systems)Energy parameters (electron-doped systems)
For electron-doped cuprates, we use the second form of the Spectral Theorem to get exponentially small
terms:
For electron-doped cuprates, we use the second form of the Spectral Theorem to get exponentially small
terms:
GMFA Correlation Functions for Singlet Hopping
GMFA Correlation Functions for Singlet Hopping
GMFA Correlation Functions for Superconducting Pairing
GMFA Correlation Functions for Superconducting Pairing
V. Frequency Matrix and
Green Function in Reciprocal
Space
V. Frequency Matrix and
Green Function in Reciprocal
Space
Frequency Matrix in (q, ω)-representation (1)
Frequency Matrix in (q, ω)-representation (1)
Frequency Matrix in (q, ω)-representation (2)
Frequency Matrix in (q, ω)-representation (2)
Frequency Matrix in (q, ω)-representation (3)
Frequency Matrix in (q, ω)-representation (3)
Frequency Matrix in (q, ω)-representation (4)
Frequency Matrix in (q, ω)-representation (4)
Frequency Matrix in (q, ω)-representation (5)
Frequency Matrix in (q, ω)-representation (5)
GMFA Green function matrix in (q, ω)-representation (1)
GMFA Green function matrix in (q, ω)-representation (1)
GMFA-GF Matrix in (q, ω)-representation (2)
GMFA-GF Matrix in (q, ω)-representation (2)
GMFA Energy SpectrumGMFA Energy Spectrum
VI. DISCUSSION
VI. DISCUSSION
1.1 We considered the effective two-band Hubbard model of high-Tc superconductivity in cuprates [N.M. Plakida et al. PRB 51, 16599 (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
1.1 We considered the effective two-band Hubbard model of high-Tc superconductivity in cuprates [N.M. Plakida et al. PRB 51, 16599 (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
Thank you for your attention !
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