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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
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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
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I. Physical Motivation
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Damascelli et al., RMP, 75, 473, 2003
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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)
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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
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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).
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(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)
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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.
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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
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Previous Results of Hubbard Model Studies
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Two-subband effective Hubbard model: AFM exchange pairing
e1 i j Estimate in WCA gives for Tcex : N.M.Plakida, L.Anton, S.Adam, Gh.Adam, JETP, 97, 331 (2003)
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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)
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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)
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Dispersion Curves (a) and Spectral Functions (b)
δ=0.05 δ=0.3 N.M.Plakida, V.S. Oudovenko JETP, 104, 230 (2007)
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II. Model Hamiltonian
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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)
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Properties of Hubbard Operators
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Hubbard Operators (1)
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Hubbard Operators (2)
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Hubbard Operators (3)
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[Properties of Hubbard Operators]
End [Properties of Hubbard Operators]
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Hubbard p-Form of labels
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Hubbard 1- Forms in Hamiltonian
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The Hamiltonian in terms of Hubbard 1-forms
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Energy Parameters (1)
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Energy Parameters (2)
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Energy Parameters (3)
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Hopping contributions to the Hamiltonian in terms of Hubbard linear forms
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III. Mean Field Approximation
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Consequences of translation invariance of the spin lattice
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Mean Field Approximation
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Frequency matrix under spin reversal
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Deriving spin reversal invariance properties
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Normal one-site statistical averages
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Anomalous one-site statistical averages
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Two-site statistical averages
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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.
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Frequency matrix in (r,ω)-representation
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Frequency Matrix in (r,ω)-representation
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The Normal Hopping Matrix
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Consequences of spin reversal invariance
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The Anomalous Hopping Matrix
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IV. Reduction of Correlation Order of Processes Involving Singlets
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Energy parameters (hole-doped cuprates)
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For hole-doped cuprates, the Spectral Theorem gives:
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Result of reduction of correlation order
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Energy parameters (electron-doped systems)
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For electron-doped cuprates, we use the second form of the Spectral Theorem to get exponentially small terms:
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GMFA Correlation Functions for Singlet Hopping
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GMFA Correlation Functions for Superconducting Pairing
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V. Frequency Matrix and Green Function in Reciprocal Space
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Frequency Matrix in (q, ω)-representation (1)
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Frequency Matrix in (q, ω)-representation (2)
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Frequency Matrix in (q, ω)-representation (3)
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Frequency Matrix in (q, ω)-representation (4)
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Frequency Matrix in (q, ω)-representation (5)
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GMFA Green function matrix in (q, ω)-representation (1)
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GMFA-GF Matrix in (q, ω)-representation (2)
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GMFA Energy Spectrum
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VI. DISCUSSION
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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
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Thank you for your attention !
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