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Mean field Green function Mean field Green function solution of the two-band solution of the two-band
Hubbard model in cupratesHubbard model in cuprates
Gh. Adam, S. AdamLaboratory of Information Technologies
JINR-Dubna
Mathematical Modeling and Computational PhysicsDubna, July 7 - 11, 2009
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).
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
I. Main Features of Cuprate
Superconductors
I. Main Features of Cuprate
Superconductors
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.
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
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
From: http://en.wikipedia.org/wiki/High-temperature_superconductivity
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
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.
• 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
Hubbard Operators: Hubbard Operators: at lattice site at lattice site iiHubbard Operators: Hubbard Operators: at lattice site at lattice site ii
Hubbard Operators:Hubbard Operators: algebraalgebraHubbard Operators:Hubbard Operators: algebraalgebra
(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 lattice plane to effective spin latticeFrom CuOFrom CuO22 plane to effective spin lattice plane to effective spin lattice
HubbardHubbard 11-Forms Forms in in HamiltonianHamiltonianHubbardHubbard 11-Forms Forms in in HamiltonianHamiltonian
II. Model Hamiltoni
an
II. Model Hamiltoni
an
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
Gh. Adam, S. Adam, J.Phys.A: Math. & Theor., 40, 11205 (2007)
Standard HamiltonianStandard Hamiltonianin terms of Hubbard 1-formsin terms of Hubbard 1-forms
Standard HamiltonianStandard Hamiltonianin terms of Hubbard 1-formsin terms of Hubbard 1-forms
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 factorLocally manifest Hermitian HamiltonianLocally manifest Hermitian Hamiltonian with hopping boundary condition factorwith hopping boundary condition factor
III. Rigorous Mean Field Solution of
Green Function Matrix
III. Rigorous Mean Field Solution of
Green Function Matrix
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
Definition of Definition of Green Function Green Function MatrixMatrixDefinition of Definition of Green Function Green Function MatrixMatrix
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
Fundamental anticommutatorsFundamental anticommutators
Deriving spin reversal invariance propertiesDeriving spin reversal invariance properties
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 predefinedspin projection. The total number of spin states equals 2.Appropriate description of effects coming from a given subbandasks for use of the specific particle number operator.
At a given lattice site i, there is a single spin state of predefinedspin projection. The total number of spin states equals 2.Appropriate description of effects coming from a given subbandasks for use of the specific particle number operator.
Average occupation numbersAverage occupation numbers
One-site singlet processesOne-site singlet processes
Normal one-site hopping processesNormal one-site hopping processes
Anomalous one-site hopping processesAnomalous one-site hopping processes
Charge-spin separation for two-site normal processes
Charge-spin separation for two-site normal processes
Charge-charge correlation mechanism of the superconducting pairing
Charge-charge correlation mechanism of the superconducting pairing
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
Localized Cooper pairs (1)
Localized Cooper pairs (1)
Localized Cooper pairs (2)
Localized Cooper pairs (2)
GMFA Correlation Functions for Singlet Hopping
GMFA Correlation Functions for Singlet Hopping
V. Setting the boundary condition
factor
V. Setting the boundary condition
factorIn 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.
Green function matrix in Green function matrix in ((qq, , ωω)-)-representationrepresentationGreen function matrix in Green function matrix in ((qq, , ωω)-)-representationrepresentation
Energy matrix in Energy matrix in ((qq, , ωω)-)-representationrepresentationEnergy matrix in Energy matrix in ((qq, , ωω)-)-representationrepresentation
VI. Energy SpectrumVI. Energy Spectrum
Hybridization of normal state energy levels preserves the center of gravityof the unhybridized levels.
Hybridization of superconducting state energy levels displaces the centerof gravity of the unhybridized normal levels. The whole spectrum is displaced towards lower frequencies.
Hybridization of normal state energy levels preserves the center of gravityof the unhybridized levels.
Hybridization of superconducting state energy levels displaces the centerof gravity of the unhybridized normal levels. The whole spectrum is displaced towards lower frequencies.
Normal State GMFA Energy SpectrumNormal State GMFA Energy Spectrum
Superconducting GMFA Energy Spectrum: Secular Equation
Superconducting GMFA Energy Spectrum: Secular Equation
Superconducting GMFA Energy Spectrum: Hybridization
Superconducting GMFA Energy Spectrum: Hybridization
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-two-dimensional two-band effective Hubbard modeldimensional two-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 to by 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 substantially substantially 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-charge charge-charge (i.e.,(i.e., boson-boson)boson-boson) interactions interactions involving singlet singlet destruction/creationdestruction/creation processes and the surrounding charge surrounding charge densitydensity.
Formulation of the starting hypothesis of the two-two-dimensional two-band effective Hubbard modeldimensional two-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 to by 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 substantially substantially 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-charge charge-charge (i.e.,(i.e., boson-boson)boson-boson) interactions interactions involving singlet singlet destruction/creationdestruction/creation processes and the surrounding charge surrounding charge densitydensity.
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 pairing anomalous boson-boson pairing may be consistently consistently reformulated in terms of reformulated in terms of quasi-localized Cooper pairsquasi-localized Cooper pairs in the in the direct crystal space,direct crystal space, both for hole-doped and electron-doped cuprates. The two-sitetwo-site expressions recover the results of the recover the results of the t-J modelt-J model.
This points to the occurrence of a robust robust ddxx22-y-y22 pairingpairing both in hole-doped and electron-doped cuprates.In orthorhombic cuprates (like Y123), a small s-type small s-type admixtureadmixture 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 superconducting state state points to an overall shift of the energy levels an overall shift of the energy levels.. Hence this state is reached as a result of the minimization of the kinetic the minimization of the kinetic energyenergy of the system, in agreement with ARPES data [Molegraaf et al. Science 2002].
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