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

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