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: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).

OUTLINEI. Main Features of Cuprate

SuperconductorsII. Model HamiltonianIII. Rigorous Mean Field Solution of

Green Function MatrixIV. Reduction of Correlation Order of

Processes Involving SingletsV. Fixing the boundary condition factorVI. Energy Spectrum

I. Main Features of Cuprate

Superconductors

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 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. 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

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 Families

From: http://en.wikipedia.org/wiki/High-temperature_superconductivity

SchematicSchematic (δ,T)(δ,T) Phase Diagrams for Cuprate FamiliesPhase Diagrams for Cuprate Families

Input abstractions, concepts, factsBesides 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).Hubbard operator:

Spin lattice states:

Doped effective spin latticeDoped effective spin lattice

Hubbard Operators: Hubbard Operators: at lattice site at lattice site ii

Hubbard 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 lattice

HubbardHubbard 11-Forms Forms in in HamiltonianHamiltonian

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 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

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

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

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

Definition of Definition of Green Function Green Function MatrixMatrix

Consequences of translation invariance of the spin lattice

Mean Field Approximation

Frequency matrix under spin reversal

Fundamental anticommutators

Deriving spin reversal invariance properties

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.

Average occupation numbers

One-site singlet processes

Normal one-site hopping processes

Anomalous one-site hopping processes

Charge-spin separation for two-site normal processes

Charge-charge correlation mechanism of the superconducting pairing

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

Localized Cooper pairs (1)

Localized Cooper pairs (2)

GMFA Correlation Functions for Singlet Hopping

V. Setting the boundary condition

factorIn the model Hamiltonian, the boundary condition factor valueρ = χ1χ2

results in finite energy hopping effects over the whole range of thedoping δ both for hole-doped and electron-doped cuprates.

Green function matrix in Green function matrix in ((qq, , ωω)-)-representationrepresentation

Energy matrix in Energy matrix in ((qq, , ωω)-)-representationrepresentation

VI. 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.

Normal State GMFA Energy Spectrum

Superconducting GMFA Energy Spectrum: Secular Equation

Superconducting GMFA Energy Spectrum: Hybridization

Main 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.

Main 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].

Thank you for your attention !