Mean field Green function solution of the two-band Hubbard model in cuprates

1. Mean field Green functionMean field Green function
solution of the two-bandsolution of the two-band
Hubbard model in cupratesHubbard model in cuprates
Gh. Adam, S. Adam
Laboratory of Information Technologies
JINR-Dubna
Mathematical Modeling and Computational Physics
Dubna, July 7 - 11, 2009

2. Selected references:Selected references:Selected references:Selected references:
General References on Two-Band Hubbard Model:General References on Two-Band Hubbard Model:
•N.M. Plakida, R. Hayn, J.-L. Richard, Phys. Rev. B, 51, 16599 (1995).
•N.M. Plakida, Physica C 282-287, 1737 (1997).
•N.M. Plakida, L. Anton, S. Adam, Gh. Adam, ZhETF 124, 367 (2003);
English transl.: JETP 97, 331 (2003).
•N.M. Plakida, V.S. Oudovenko, JETP 104, 230 (2007).
•N.M. Plakida “High-Temperature Superconductors. Experiment, Theory,
and Application”, Springer, 1985; 2-nd ed., to be publ.
General References on Two-Band Hubbard Model:General References on Two-Band Hubbard Model:
•N.M. Plakida, R. Hayn, J.-L. Richard, Phys. Rev. B, 51, 16599 (1995).
•N.M. Plakida, Physica C 282-287, 1737 (1997).
•N.M. Plakida, L. Anton, S. Adam, Gh. Adam, ZhETF 124, 367 (2003);
English transl.: JETP 97, 331 (2003).
•N.M. Plakida, V.S. Oudovenko, JETP 104, 230 (2007).
•N.M. Plakida “High-Temperature Superconductors. Experiment, Theory,
and Application”, Springer, 1985; 2-nd ed., to be publ.
Results on Two-Band Hubbard Model following from system symmetryResults on Two-Band Hubbard Model following from system symmetry
•Gh. Adam, S. Adam, Rigorous derivation of the mean field Green
functions of the two-band Hubbard model of superconductivity,
J.Phys.A: Mathematical and Theoretical Vol.40, 11205-11219 (2007).
•Gh. Adam, S. Adam, Separation of the spin-charge correlations in the
two-band Hubbard model of high-Tc superconductivity,
J.Optoelectronics Adv. Materials, Vol.10, 1666-1670 (2008).
S. Adam, Gh. Adam, Features of high-Tc superconducting phase
transitions in cuprates, Romanian J.Phys., Vol.53, 993-999 (2008).

Gh. Adam, S. Adam, Finiteness of the hopping induced energy
corrections in cuprates, Romanian J.Phys., Vol. 54, No. 9-10 (2009).
Results on Two-Band Hubbard Model following from system symmetryResults on Two-Band Hubbard Model following from system symmetry
•Gh. Adam, S. Adam, Rigorous derivation of the mean field Green
functions of the two-band Hubbard model of superconductivity,
J.Phys.A: Mathematical and Theoretical Vol.40, 11205-11219 (2007).
•Gh. Adam, S. Adam, Separation of the spin-charge correlations in the
two-band Hubbard model of high-Tc superconductivity,
J.Optoelectronics Adv. Materials, Vol.10, 1666-1670 (2008).
S. Adam, Gh. Adam, Features of high-Tc superconducting phase
transitions in cuprates, Romanian J.Phys., Vol.53, 993-999 (2008).

Gh. Adam, S. Adam, Finiteness of the hopping induced energy
corrections in cuprates, Romanian J.Phys., Vol. 54, No. 9-10 (2009).

3. OUTLINEOUTLINE
I. Main Features of Cuprate Superconductors
II. Model Hamiltonian
III. Rigorous Mean Field Solution of Green
Function Matrix
IV. Reduction of Correlation Order of
Processes Involving Singlets
V. Fixing the boundary condition factor
VI. Energy Spectrum
I. Main Features of Cuprate Superconductors
II. Model Hamiltonian
III. Rigorous Mean Field Solution of Green
Function Matrix
IV. Reduction of Correlation Order of
Processes Involving Singlets
V. Fixing the boundary condition factor
VI. Energy Spectrum

4. I. Main Features of
Cuprate
Superconductors
I. Main Features of
Cuprate
Superconductors

5.  Basic Principles of Theoretical Description of Cuprates Basic Principles of Theoretical Description of Cuprates
• The high critical temperature superconductivity in cuprates is still a
puzzle of the today solid state physics, in spite of the unprecedented
wave of interest & number of publications (> 105
)
• The high critical temperature superconductivity in cuprates is still a
puzzle of the today solid state physics, in spite of the unprecedented
wave of interest & number of publications (> 105
)
• The two-band Hubbard model provides a description of it based on
four basic principlesfour basic principles:
• The two-band Hubbard model provides a description of it based on
four basic principlesfour basic principles:
(1)(1) Deciding role of the experimentDeciding role of the experiment. The derivation of reliable
experimental data on various cuprate properties asks for
manufacturing high quality samples, performing high-precision
measurements, by adequate experimental methods.
(2)(2) Hierarchical orderingHierarchical ordering of the interactions inferred from data.
(3)(3) Derivation of the simplest model Hamiltonianmodel Hamiltonian following from the
Weiss principle, i.e., hierarchical implementation into the model of
the various interactions.
(4)(4) Mathematical solutionMathematical solution by right quantum statistical methods which
secure rigorous implementation of the existing physical symmetries
and observe the principles of mathematical consistency & simplicity.
(1)(1) Deciding role of the experimentDeciding role of the experiment. The derivation of reliable
experimental data on various cuprate properties asks for
manufacturing high quality samples, performing high-precision
measurements, by adequate experimental methods.
(2)(2) Hierarchical orderingHierarchical ordering of the interactions inferred from data.
(3)(3) Derivation of the simplest model Hamiltonianmodel Hamiltonian following from the
Weiss principle, i.e., hierarchical implementation into the model of
the various interactions.
(4)(4) Mathematical solutionMathematical solution by right quantum statistical methods which
secure rigorous implementation of the existing physical symmetries
and observe the principles of mathematical consistency & simplicity.

6. 1. Data: Crystal structure characterization (layered ternary perovskites).
═> Effective 2D model for CuO2 plane.
2. Data: Existence of Fermi surface.
═> Energy bands lying at or near Fermi level are to be retained.
3. Data: Charge-transfer insulator nature of cuprates.
U > Δ > W ═> hybridization results in Zhang-Rice singlet subband.
═> ZR singlet and UH subbands enter simplest model.
Δ ~ 2W ═> model to be developed in the strong correlation limit.
4. Data: Tightly bound electrons in metallic state.
═>Low density hopping conduction consisting of
both fermion and boson (singlet) carriers.
═> Need of Hubbard operatorHubbard operator description of system states.
1. Data: Cuprate families characterized by specific stoichiometric
reference structuresreference structures doped with either holes or electrons.
═> The doping parameter δ is essential; (δ, T) phase diagrams.
1. Data: Crystal structure characterization (layered ternary perovskites).
═> Effective 2D model for CuO2 plane.
2. Data: Existence of Fermi surface.
═> Energy bands lying at or near Fermi level are to be retained.
3. Data: Charge-transfer insulator nature of cuprates.
U > Δ > W ═> hybridization results in Zhang-Rice singlet subband.
═> ZR singlet and UH subbands enter simplest model.
Δ ~ 2W ═> model to be developed in the strong correlation limit.
4. Data: Tightly bound electrons in metallic state.
═>Low density hopping conduction consisting of
both fermion and boson (singlet) carriers.
═> Need of Hubbard operatorHubbard operator description of system states.
1. Data: Cuprate families characterized by specific stoichiometric
reference structuresreference structures doped with either holes or electrons.
═> The doping parameter δ is essential; (δ, T) phase diagrams.
 Experimental Input to Theoretical Model Experimental Input to Theoretical Model

7. A.Erb et al.: http://www.wmi.badw-muenchen.de/FG538/projects/P4_crystal_growth/index.htm
SchematicSchematic (δ,T)(δ,T) Phase Diagrams for Cuprate FamiliesPhase Diagrams for Cuprate FamiliesSchematicSchematic (δ,T)(δ,T) Phase Diagrams for Cuprate FamiliesPhase Diagrams for Cuprate Families

8. From: http://en.wikipedia.org/wiki/High-temperature_superconductivi
SchematicSchematic (δ,T)(δ,T) Phase Diagrams for Cuprate FamiliesPhase Diagrams for Cuprate FamiliesSchematicSchematic (δ,T)(δ,T) Phase Diagrams for Cuprate FamiliesPhase Diagrams for Cuprate Families

9.  Input abstractions, concepts, facts Input abstractions, concepts, facts
Besides the straightforward inferences following from the experiment,
a number of input items need consideration.
1. Abstraction of the physical CuO2 plane with doped electron states.
═> Doped effective spin lattice.
2. Concept: Global description of the hopping conduction around a spin
lattice site.
═> Hubbard 1-forms.
3. Fact: The hopping induced energy correction effects are finite over
the whole available range of the doping parameter δ.
═> Appropriate boundary conditions are to be imposed in the limit
of vanishing doping.
Besides the straightforward inferences following from the experiment,
a number of input items need consideration.
1. Abstraction of the physical CuO2 plane with doped electron states.
═> Doped effective spin lattice.
2. Concept: Global description of the hopping conduction around a spin
lattice site.
═> Hubbard 1-forms.
3. Fact: The hopping induced energy correction effects are finite over
the whole available range of the doping parameter δ.
═> Appropriate boundary conditions are to be imposed in the limit
of vanishing doping.

10. • One-to-one mapping from the copper sites inside CuO2 plane to the
spins of the effective spin lattice.
═> Spin lattice constants equal ax, ay, the CuO2 lattice constants.
═> Antiferromagnetic spin ordering at zero doping.
• Doping of electron states inside CuO2 plane <═> creation of defects
inside the spin lattice by spin vacancies and/or singlet states.
• Hopping conductivity inside spin lattice: a consequence of doping.
═> It may consist either of single spin hopping (fermionic
conductivity) or singlet hopping (bosonic conductivity).
• One-to-one mapping from the copper sites inside CuO2 plane to the
spins of the effective spin lattice.
═> Spin lattice constants equal ax, ay, the CuO2 lattice constants.
═> Antiferromagnetic spin ordering at zero doping.
• Doping of electron states inside CuO2 plane <═> creation of defects
inside the spin lattice by spin vacancies and/or singlet states.
• Hopping conductivity inside spin lattice: a consequence of doping.
═> It may consist either of single spin hopping (fermionic
conductivity) or singlet hopping (bosonic conductivity).
Hubbard operator:
Spin lattice states:
Doped effective spin latticeDoped effective spin latticeDoped effective spin latticeDoped effective spin lattice

11. Hubbard Operators:Hubbard Operators: at lattice siteat lattice site iiHubbard Operators:Hubbard Operators: at lattice siteat lattice site ii

12. Hubbard Operators:Hubbard Operators: algebraalgebraHubbard Operators:Hubbard Operators: algebraalgebra

13. (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
i
j
From CuOFrom CuO22 plane to effective spin latticeplane to effective spin latticeFrom CuOFrom CuO22 plane to effective spin latticeplane to effective spin lattice

14. HubbardHubbard 11-FormsForms inin HamiltonianHamiltonianHubbardHubbard 11-FormsForms inin HamiltonianHamiltonian

15. II. Model
Hamiltonian
II. Model
Hamiltonian

16. 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)
Standard HamiltonianStandard HamiltonianStandard HamiltonianStandard Hamiltonian

17. Gh. Adam, S. Adam, J.Phys.A: Math. & Theor., 40, 11205 (2007)
Standard HamiltonianStandard Hamiltonian
in terms of Hubbard 1-formsin terms of Hubbard 1-forms
Standard HamiltonianStandard Hamiltonian
in terms of Hubbard 1-formsin terms of Hubbard 1-forms

18. Gh. Adam, S. Adam, Romanian J. Phys., 54, No.9-10 (2009)
Locally manifest Hermitian HamiltonianLocally manifest Hermitian Hamiltonian
with hopping boundary condition factorwith hopping boundary condition factor
Locally manifest Hermitian HamiltonianLocally manifest Hermitian Hamiltonian
with hopping boundary condition factorwith hopping boundary condition factor

19. III. Rigorous
Mean Field
Solution of Green
Function Matrix
III. Rigorous
Mean Field
Solution of Green
Function Matrix

20. RepresentationsRepresentations
(r, t)
[space-time]
F.T. F.T.
(r, ω)
[space-energy]
F.T. F.T.
(q, ω)
[momentum-energy]
F.T. = Fourier transform
RepresentationsRepresentations
(r, t)
[space-time]
F.T. F.T.
(r, ω)
[space-energy]
F.T. F.T.
(q, ω)
[momentum-energy]
F.T. = Fourier transform
 Direct & Dual Formulations of Green Functions (GF) Direct & Dual Formulations of Green Functions (GF)
ActionsActions
• Retarded/Advanced GF definitions
• GF differential equations of motion
• Splitting higher order correlation functions
• GF algebraic equations of motion
• Analytic extensions in complex energy plane.
Unique GF in complex plane.
• Statistical average calculations from spectral
theorems
• Compact functional GF expressions
• Equations for the energy spectra
• Statistical average calculations from spectral
theorems
• Spectral distributions inside Brillouin zone
ActionsActions
• Retarded/Advanced GF definitions
• GF differential equations of motion
• Splitting higher order correlation functions
• GF algebraic equations of motion
• Analytic extensions in complex energy plane.
Unique GF in complex plane.
• Statistical average calculations from spectral
theorems
• Compact functional GF expressions
• Equations for the energy spectra
• Statistical average calculations from spectral
theorems
• Spectral distributions inside Brillouin zone

21. Definition ofDefinition of Green FunctionGreen Function MatrixMatrixDefinition ofDefinition of Green FunctionGreen Function MatrixMatrix

22. Consequences of translation invariance
of the spin lattice
Consequences of translation invariance
of the spin lattice

24. Mean Field ApproximationMean Field Approximation

25. Frequency matrix under spin reversalFrequency matrix under spin reversal

26. Fundamental anticommutatorsFundamental anticommutators

29. Deriving spin reversal invariance propertiesDeriving spin reversal invariance properties

30. Appropriate particle number operators describe
correlations coming from each subband
Appropriate particle number operators describe
correlations coming from each subband
At a given lattice site i, there is a single spin state of predefined
spin projection. The total number of spin states equals 2.
Appropriate description of effects coming from a given subband
asks for use of the specific particle number operator.
At a given lattice site i, there is a single spin state of predefined
spin projection. The total number of spin states equals 2.
Appropriate description of effects coming from a given subband
asks for use of the specific particle number operator.

31. Average occupation numbersAverage occupation numbers

32. One-site singlet processesOne-site singlet processes

33. Normal one-site hopping processesNormal one-site hopping processes

34. Anomalous one-site hopping processesAnomalous one-site hopping processes

35. Charge-spin separation for two-site
normal processes
Charge-spin separation for two-site
normal processes

36. Charge-charge correlation mechanism of the
superconducting pairing
Charge-charge correlation mechanism of the
superconducting pairing

37. IV. Reduction of
Correlation Order
of Processes
Involving Singlets
IV. Reduction of
Correlation Order
of Processes
Involving Singlets
• Spectral theorem for the statistical averages of interest
• Equations of motion for retarded & advanced integrand Green Functions
• Neglect of exponentially small terms
• Principal part integrals yield relevant contributions to averages
• Spectral theorem for the statistical averages of interest
• Equations of motion for retarded & advanced integrand Green Functions
• Neglect of exponentially small terms
• Principal part integrals yield relevant contributions to averages

38. Localized Cooper pairs (1)Localized Cooper pairs (1)

39. Localized Cooper pairs (2)Localized Cooper pairs (2)

40. GMFA Correlation Functions for
Singlet Hopping
GMFA Correlation Functions for
Singlet Hopping

41. V. Setting the
boundary
condition factor
V. Setting the
boundary
condition factor
In the model Hamiltonian, the boundary condition factor value
ρ = χ1χ2
results in finite energy hopping effects over the whole range of the
doping δ both for hole-doped and electron-doped cuprates.
In the model Hamiltonian, the boundary condition factor value
ρ = χ1χ2
results in finite energy hopping effects over the whole range of the
doping δ both for hole-doped and electron-doped cuprates.

42. Green function matrix inGreen function matrix in ((qq,, ωω)-)-representationrepresentationGreen function matrix inGreen function matrix in ((qq,, ωω)-)-representationrepresentation

43. Energy matrix inEnergy matrix in ((qq,, ωω)-)-representationrepresentationEnergy matrix inEnergy matrix in ((qq,, ωω)-)-representationrepresentation

44. VI. Energy
Spectrum
VI. Energy
Spectrum
Hybridization of normal state energy levels preserves the center of gravity
of the unhybridized levels.
Hybridization of superconducting state energy levels displaces the center
of gravity of the unhybridized normal levels.
The whole spectrum is displaced towards lower frequencies.
Hybridization of normal state energy levels preserves the center of gravity
of the unhybridized levels.
Hybridization of superconducting state energy levels displaces the center
of gravity of the unhybridized normal levels.
The whole spectrum is displaced towards lower frequencies.

45. Normal State GMFA Energy SpectrumNormal State GMFA Energy Spectrum

46. Superconducting GMFA Energy Spectrum:
Secular Equation
Superconducting GMFA Energy Spectrum:
Secular Equation

47. Superconducting GMFA Energy Spectrum:
Hybridization
Superconducting GMFA Energy Spectrum:
Hybridization

48. Main new results in this researchMain new results in this researchMain new results in this researchMain new results in this research
 Formulation of the starting hypothesis of the two-dimensionaltwo-dimensional
two-band effective Hubbard modeltwo-band effective Hubbard model, allowed the definition of the
model Hamiltonianmodel Hamiltonian as a sum
H = H0 + χ1χ2 Hh
which covers consistently the whole doping range in the (δ,Τ)-phase
diagrams of both hole-doped (χ2 << 1) and electron-doped (χ1 << 1)
cuprates, including the reference structureincluding the reference structure at δ = 0.
The spin-charge separation,spin-charge separation, conjectured by P.W. Anderson toby P.W. Anderson to
happen in cuprates,happen in cuprates, is rigorously observedis rigorously observed under the existence of the
Fermi surfaceFermi surface in these compounds.
Remark: This result differsdiffers substantiallysubstantially from Anderson’s “spin protectorate”
scenario which denies the existence of the Fermi surface.
 The static exchange superconducting mechanismThe static exchange superconducting mechanism is intimately
related to the singlet conductionsinglet conduction. It stems from charge-chargecharge-charge (i.e.,(i.e.,
boson-boson)boson-boson) interactionsinteractions involving singlet destruction/creationsinglet destruction/creation
processes and the surrounding charge densitysurrounding charge density.
 Formulation of the starting hypothesis of the two-dimensionaltwo-dimensional
two-band effective Hubbard modeltwo-band effective Hubbard model, allowed the definition of the
model Hamiltonianmodel Hamiltonian as a sum
H = H0 + χ1χ2 Hh
which covers consistently the whole doping range in the (δ,Τ)-phase
diagrams of both hole-doped (χ2 << 1) and electron-doped (χ1 << 1)
cuprates, including the reference structureincluding the reference structure at δ = 0.
The spin-charge separation,spin-charge separation, conjectured by P.W. Anderson toby P.W. Anderson to
happen in cuprates,happen in cuprates, is rigorously observedis rigorously observed under the existence of the
Fermi surfaceFermi surface in these compounds.
Remark: This result differsdiffers substantiallysubstantially from Anderson’s “spin protectorate”
scenario which denies the existence of the Fermi surface.
 The static exchange superconducting mechanismThe static exchange superconducting mechanism is intimately
related to the singlet conductionsinglet conduction. It stems from charge-chargecharge-charge (i.e.,(i.e.,
boson-boson)boson-boson) interactionsinteractions involving singlet destruction/creationsinglet destruction/creation
processes and the surrounding charge densitysurrounding charge density.

49. Main new results in this researchMain new results in this researchMain new results in this researchMain new results in this research
 The anomalous boson-boson pairinganomalous boson-boson pairing may be consistentlyconsistently
reformulated in terms ofreformulated in terms of quasi-localized Cooper pairsquasi-localized Cooper pairs in the directin the direct
crystal space,crystal space, both for hole-doped and electron-doped cuprates.
The two-sitetwo-site expressions recover the results of the t-J modelrecover the results of the t-J model.
This points to the occurrence of a robustrobust ddxx22
-y-y22 pairingpairing both in
hole-doped and electron-doped cuprates.
In orthorhombic cuprates (like Y123), a small s-type admixturesmall s-type admixture is
predicted to occur, in qualitative agreement with the phase sensitive
experiments [Kirtley, Tsuei et al., Nature Physics 2006]
 The energy spectrum calculationenergy spectrum calculation of the superconducting statesuperconducting state
points to an overall shift of the energy levelsan overall shift of the energy levels.. Hence this state is
reached as a result of the minimization of the kinetic energythe minimization of the kinetic energy of the
system, in agreement with ARPES data [Molegraaf et al. Science 2002].

50. Thank you for your
attention !