Transcript Hubbard model

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
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.
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-Tc 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).
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

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 principles:
(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 of the interactions inferred from data.
(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 by right quantum statistical methods which
secure rigorous implementation of the existing physical symmetries
and observe the principles of mathematical consistency & simplicity.
 Experimental Input to Theoretical Model
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 operator description of system states.
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.
Schematic
(δ,T) Phase Diagrams for Cuprate Families
A.Erb et al.: http://www.wmi.badw-muenchen.de/FG538/projects/P4_crystal_growth/index.htm
Schematic
(δ,T) Phase Diagrams for Cuprate Families
From: http://en.wikipedia.org/wiki/High-temperature_superconductivity
 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.
Doped effective spin lattice
• 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:
Hubbard Operators: at lattice site i
Hubbard Operators: algebra
From CuO2 plane to effective spin lattice
i
j
(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
Hubbard 1-Forms in Hamiltonian
II. Model
Hamiltonian
Standard 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)
Standard Hamiltonian
in terms of Hubbard 1-forms
Gh. Adam, S. Adam, J.Phys.A: Math. & Theor., 40, 11205 (2007)
Locally manifest Hermitian Hamiltonian
with hopping boundary condition factor
Gh. Adam, S. Adam, Romanian J. Phys., 54, No.9-10 (2009)
III. Rigorous
Mean Field
Solution of Green
Function Matrix

Direct & Dual Formulations of Green Functions (GF)
Representations
(r, t)
[space-time]
F.T.
• Retarded/Advanced GF definitions
• GF differential equations of motion
• Splitting higher order correlation functions
F.T.
(r, ω)
[space-energy]
F.T.
Actions
F.T.
(q, ω)
[momentum-energy]
F.T. = Fourier transform
• 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 Green Function Matrix
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 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.
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 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.
Green function matrix in (q, ω)-representation
Energy matrix in (q, ω)-representation
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.
Normal State GMFA Energy Spectrum
Superconducting GMFA Energy Spectrum:
Secular Equation
Superconducting GMFA Energy Spectrum:
Hybridization
Main new results in this research
 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 = 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 structure at δ = 0.
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.
Remark: This result differs substantially from Anderson’s “spin protectorate”
scenario which denies the existence of the Fermi surface.
 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.
Main new results in this research
 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.
This points to the occurrence of a robust dx2-y2 pairing both in
hole-doped and electron-doped cuprates.
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]
 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].
Thank you for your attention !