1. Mean field Green function solution of the two-band Hubbard 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: 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: 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 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-T c superconductivity, J.Optoelectronics Adv. Materials, Vol.10, 1666-1670 (2008). S. Adam, Gh. Adam, Features of high-T c 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 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-T c superconductivity, J.Optoelectronics Adv. Materials, Vol.10, 1666-1670 (2008). S. Adam, Gh. Adam, Features of high-T c 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. OUTLINE 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

5.  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 (> 10 5 ) four basic principles The two-band Hubbard model provides a description of it based on four basic principles: (1) Deciding role of the experiment (1) Deciding 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) Hierarchical ordering (2) Hierarchical ordering of the interactions inferred from data. (3) model Hamiltonian (3) Derivation of the simplest model Hamiltonian following from the Weiss principle, i.e., hierarchical implementation into the model of the various interactions. (4) Mathematical solution (4) Mathematical solution by right quantum statistical methods which secure rigorous implementation of the existing physical symmetries and observe the principles of mathematical consistency & simplicity. (1) Deciding role of the experiment (1) Deciding 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) Hierarchical ordering (2) Hierarchical ordering of the interactions inferred from data. (3) model Hamiltonian (3) Derivation of the simplest model Hamiltonian following from the Weiss principle, i.e., hierarchical implementation into the model of the various interactions. (4) Mathematical solution (4) Mathematical 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 CuO 2 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. Hubbard operator ═> Need of Hubbard operator description of system states. reference structures 5.Data: Cuprate families characterized by specific stoichiometric reference 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 CuO 2 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. Hubbard operator ═> Need of Hubbard operator description of system states. reference structures 5.Data: Cuprate families characterized by specific stoichiometric reference structures doped with either holes or electrons. ═> The doping parameter δ is essential; ( δ, T ) phase diagrams.  Experimental Input to Theoretical Model

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

8. From: http://en.wikipedia.org/wiki/High-temperature_superconductivity http://en.wikipedia.org/wiki/High-temperature_superconductivity Schematic (δ,T) Phase Diagrams for Cuprate Families

9.  Input abstractions, concepts, facts Besides the straightforward inferences following from the experiment, a number of input items need consideration. 1.Abstraction of the physical CuO 2 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 CuO 2 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 CuO 2 plane to the spins of the effective spin lattice. ═> Spin lattice constants equal a x, a y, the CuO 2 lattice constants. ═> Antiferromagnetic spin ordering at zero doping. Doping of electron states inside CuO 2 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 CuO 2 plane to the spins of the effective spin lattice. ═> Spin lattice constants equal a x, a y, the CuO 2 lattice constants. ═> Antiferromagnetic spin ordering at zero doping. Doping of electron states inside CuO 2 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 lattice

11. Hubbard Operators: at lattice site i‏

12. Hubbard Operators: algebra

13. (a) Schematic representation of the cell distribution within CuO 2 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 CuO 2 plane to effective spin lattice

14. Hubbard 1 Forms in Hamiltonian Hubbard 1- Forms in Hamiltonian

15. 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 Hamiltonian

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

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

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

20. Representations (r, t) [space-time] F.T. F.T. (r, ω) [space-energy] F.T. F.T. (q, ω) [momentum-energy] F.T. = Fourier transformRepresentations (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) Actions 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 zoneActions 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 of Green Function Matrix

22. Consequences of translation invariance of the spin lattice

24. Mean Field Approximation

25. Frequency matrix under spin reversal

26. Fundamental anticommutators

29. Deriving spin reversal invariance properties

30. 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 numbers

32. One-site singlet processes

33. Normal one-site hopping processes

34. Anomalous one-site hopping processes

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

36. Charge-charge correlation mechanism of the superconducting pairing

37. 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) ‏

39. Localized Cooper pairs (2) ‏

40. GMFA Correlation Functions for Singlet Hopping

41. V. Setting the boundary condition factor In the model Hamiltonian, the boundary condition factor value ρ = χ 1 χ 2 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 = χ 1 χ 2 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 in ( q, ω )- representation

43. Energy matrix in ( q, ω )- representation

44. 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 Spectrum

46. Superconducting GMFA Energy Spectrum: Secular Equation

47. Superconducting GMFA Energy Spectrum: Hybridization

48. Main new results in this research two-dimensional two-band effective Hubbard model model Hamiltonian  Formulation of the starting hypothesis of the two-dimensional two-band effective Hubbard model, allowed the definition of the model Hamiltonian as a sum H = H 0 + χ 1 χ 2 H h including the reference structure which covers consistently the whole doping range in the (δ,Τ)- phase diagrams of both hole-doped ( χ 2  1 ) and electron-doped ( χ 1  1 ) cuprates, including the reference structure at δ = 0. spin-charge separation, by P.W. Anderson to happen in cuprates,is rigorously observed Fermi surface  The spin-charge separation, conjectured by P.W. Anderson to happen in cuprates, is rigorously observed under the existence of the Fermi surface in these compounds. differssubstantially Remark: This result differs substantially from Anderson’s “spin protectorate” scenario which denies the existence of the Fermi surface. The static exchange superconducting mechanism singlet conductioncharge-charge (i.e., boson-boson) interactionssinglet destruction/creation surrounding charge density  The static exchange superconducting mechanism is intimately related to the singlet conduction. It stems from charge-charge (i.e., boson-boson) interactions involving singlet destruction/creation processes and the surrounding charge density. two-dimensional two-band effective Hubbard model model Hamiltonian  Formulation of the starting hypothesis of the two-dimensional two-band effective Hubbard model, allowed the definition of the model Hamiltonian as a sum H = H 0 + χ 1 χ 2 H h including the reference structure which covers consistently the whole doping range in the (δ,Τ)- phase diagrams of both hole-doped ( χ 2  1 ) and electron-doped ( χ 1  1 ) cuprates, including the reference structure at δ = 0. spin-charge separation, by P.W. Anderson to happen in cuprates,is rigorously observed Fermi surface  The spin-charge separation, conjectured by P.W. Anderson to happen in cuprates, is rigorously observed under the existence of the Fermi surface in these compounds. differssubstantially Remark: This result differs substantially from Anderson’s “spin protectorate” scenario which denies the existence of the Fermi surface. The static exchange superconducting mechanism singlet conductioncharge-charge (i.e., boson-boson) interactionssinglet destruction/creation surrounding charge density  The static exchange superconducting mechanism is intimately related to the singlet conduction. It stems from charge-charge (i.e., boson-boson) interactions involving singlet destruction/creation processes and the surrounding charge density.

49. Main new results in this research anomalous boson-boson pairing consistently reformulated in terms of quasi-localized Cooper pairs in the direct crystal space, two-siterecover the results of the t-J model  The anomalous boson-boson pairing may be consistently reformulated in terms of quasi-localized Cooper pairs in the direct crystal space, both for hole-doped and electron-doped cuprates. The two-site expressions recover the results of the t-J model. robust d x 2 -y 2 pairing  This points to the occurrence of a robust d x 2 -y 2 pairing both in hole-doped and electron-doped cuprates. small s-type admixture In orthorhombic cuprates (like Y123), a small s-type admixture is predicted to occur, in qualitative agreement with the phase sensitive experiments [ Kirtley, Tsuei et al., Nature Physics 2006] energy spectrum calculationsuperconducting state an overall shift of the energy levels. the minimization of the kinetic energy  The energy spectrum calculation of the superconducting state points to an overall shift of the energy levels. Hence this state is reached as a result of the minimization of the kinetic energy of the system, in agreement with ARPES data [ Molegraaf et al. Science 2002 ].

50. Thank you for your attention !