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Correlations in quantum dots: How far can analytics go?
Vyacheslavs (Slava) Kashcheyevs Collaboration: Amnon Aharony (BGU+TAU) Ora Entin-Wohlman (BGU+TAU) Avraham Schiller (Hebrew Univ.) December 18th, 2006
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Outline Physics of the Anderson model Equations of motion
Relevant energies and regimes The plethora of methods Equations of motion Self-consistent truncation Gauging quality in known limits Anderson model for double quantum dots Charge location as a pseudo-spin Diverse results in a unified way The first is a pillar-builder. No equations One is more about the method (lot’s of equations), the other one – on results (all together)
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Quantum dots Tune: gate potentials, temperature, field…
Lead QD Lead Gates Electron gas plane GaAs AlGaAs Tune: gate potentials, temperature, field… Measure: I-V curves, conductance G… Aharonov-Bohm interferometry, dephasing, coherent state manipulation…
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Building models Quantum dots Tunneling Hamiltonian approach
Mesoscopic: many levels involved, statistical description Microscopic: few levels, individual properties Tunneling Hamiltonian approach
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Anderson model: the leads
μ Non-interacting in the sense of Fermi liquid. Continuum, equilibrium, chemical potetnial Lead Lead A set of energy levels
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Anderson model: the dot
μ Also a level. epsilon_0 >> mu – empty, does down, gets occupied, occupied. Lead QD Lead An energy level ε0
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Anderson model: the dot
ε0 1 2 n f (ε0) T μ μ Also a level. epsilon_0 >> mu – empty, does down, gets occupied, occupied. A Fermion! fermi-Dirac statistic Lead QD Lead An energy level ε0
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Anderson model: the dot
ε0 1 2 n f (ε0) T μ μ A Fermion! fermi-Dirac statistic. remember spin Lead QD Lead An energy level ε0
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Anderson model: the dot
Spin-charge separation or “Mott transition” ε0 1 2 n f (ε0) T U U μ μ μ Interactuions Interactions! Lead QD Lead An energy level ε0
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Anderson model: tunneling
ε0 1 2 n VR U VL Γ Γ μ μ Quantum fluctuations: – of charge Γ > T – of spin U > Γ > T Interactuions Lead QD Lead Tunneling Tunneling rate Γ ~ ρ|V|2
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Anderson model: spin exchange
ε0 1 2 n U μ μ Interactuions Lead QD Lead Fix the spin on the dot Opposite spin in the leads can lower energy!
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Anderson model: spin exchange
ε0 1 2 n U μ μ Virtual transition: Interactuions Lead QD Lead Fix the spin on the dot Opposite spin in the leads can lower energy!
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Anderson model: spin exchange
ε0 1 2 n U μ μ Virtual transition: Interactuions Lead QD Lead Fix the spin on the dot Same spin can’t go!
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Effective Hamiltonian: Kondo
Interactuions Lead QD Lead Ferromagnetic exchange interaction!
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Effective Hamiltonian: Kondo
<S> ½ – O(J) 1/2 ? h J Interactuions Lead QD Lead Ferromagnetic exchange interaction! ← can fix S with h >> J
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The Kondo effect <S> ½ – O(J) 1/2 ? ~ h/TK h TK J Interactuions Lead QD Lead singlet-triplet splitting |↑↓> and |↓↑> are degenerate (Sz=0 of S=0,1) except for virtual excitations ~ J !
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Outline Physics of the Anderson model Equations of motion
Relevant energies and regimes The plethora of methods Equations of motion Self-consistent truncation Gauging quality in known limits Anderson model for double quantum dots Charge location as a pseudo-spin Diverse results in a unified way The first is a pillar-builder. No equations One is more about the method (lot’s of equations), the other one – on results (all together)
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Methods Perturbation theory (PT) in Γ, in U
in U – regular & systematic; not good for U>> Γ. in Γ – breaks down at resonances & in Kondo regime Fermi liquid: good at T<TK, exact sum rules for T=0 Equations of motion (EOM) for Green functions exact for U; a low order (Hartree mean field) gives local moment as good as PT when PT is valid can we get Kondo at higher orders? Renormalization Group perturbative (in Γ) RG => Kondo Hamiltonian + PM scaling perturbative (in U) RG => functional RG (semi-analytic) Wilson’s Numerical RG – high accuracy, but numerics only Bethe ansatz: exploits integrability exact (!) solution, many analytic results integrability condition too restrictive, finite very T laborious
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Outline Physics of the Anderson model Equations of motion
Relevant energies and regimes The plethora of methods Equations of motion Self-consistent truncation Gauging quality in known limits Anderson model for double quantum dots Charge location as a pseudo-spin Diverse results in a unified way The first is a pillar-builder. No equations One is more about the method (lot’s of equations), the other one – on results (all together)
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The Green functions Retarded Advanced Spectral function Zubarev (1960)
step function Retarded Advanced Spectral function grand canonical
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Dot’s GF Spectral function → Density of states
Occupation number → Local charge & spin Friedel-Langreth sum rule (T=0) conductance at T=0 is proportional ~ ρ(μ)!
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Equations of motion Example: 1st equation for
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spin-flip excitations
Full solution for U=0 Lead self-energy function ε0 +U ε0 ω=0 Fermi hole excitations electron excitations spin-flip excitations Lorenzian DOS bandwidth D Γ Large U should bring
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Full hierarchy …
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A general term Dworin (1967) m = 0,1, 2… lead operators n = 0 – 3 dot operators Contributes to a least m-th order in Vk and (n+m–1)/2-th order in U ! Certain order of EOM truncation ↔ certain order of perturbation theory!
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Decoupling
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Demand full self-consistency
Decoupling “D.C.Mattis scheme”: Theumann (1969) Use mean-field for at most 1 dot operator: Use values Meir, Wigreen, Lee (1991) Linear = easy to solve Fails at low T – no Kondo Demand full self-consistency Significant improvement Hard-to-solve non-linear integral eqs.
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The self-consistent equations
Zeeman splitting Level position The only input parameters Self-consistent functions:
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EOMs: How to solve? In general, iterative numerical solution
Two analytically solvable cases: and wide band limit: explicit non-trivial solution particle-hole symmetry point : break down of the approximation
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Outline Physics of the Anderson model Equations of motion
Relevant energies and regimes The plethora of methods Equations of motion Self-consistent truncation Gauging quality in known limits Anderson model for double quantum dots Charge location as a pseudo-spin Diverse results in a unified way The first is a pillar-builder. No equations One is more about the method (lot’s of equations), the other one – on results (all together)
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EOMs: Results Ed / Γ Zero temperature Zero magnetic field & wide band
Energy ω/Γ Fermi Zero temperature Zero magnetic field & wide band Level renormalization Changing Ed/Γ Looking at DOS: even odd
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Results: occupation numbers
Compare to perturbation theory Compare to Bethe ansatz Gefen & Kőnig (2005) Wiegmann & Tsvelik (1983) Better than 3% accuracy!
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Check: Friedel-Langreth sums
No quasi-particle damping at the Fermi surface: Fermi sphere volume conservation Good – for nearly empty dot Broken – in the Kondo valley
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Results: melting of the peak
2e2/h conduct. At small T and near Fermi energy, parameters in the solution combine as Smaller than the true Kondo T: ~ 1/log2(T/TK) DOS at the Fermi energy scales with T/TK* Experiment: van der Wiel et al., Science 289, 2105 (2000)
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EOMs: conclusions “Physics repeats itself with a period of T ≈ 30 years” – © OEW Non-trivial results require non-trivial effort … and even then they may disappoint someone’s expectations
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Outline Physics of the Anderson model Equations of motion
Relevant energies and regimes The plethora of methods Equations of motion Self-consistent truncation Gauging quality in known limits Anderson model for double quantum dots Charge location as a pseudo-spin Diverse results in a unified way The first is a pillar-builder. No equations One is more about the method (lot’s of equations), the other one – on results (all together)
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Double dots: a minimal model
Two orbital levels Two leads Inter-dot repulsion U tunneling b Aharonov-Bohm flux (wide band)
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Interesting properties
Charge oscillations / population switching Transmission zeros / phase lapses “Correlation-induced” resonances Konig & Gefen PRB 71 (2005) [PT] Sindel, Silva, Oreg & von Delft PRB 72 (2005) [NRG & HF] Meden & Marquardt PRL (2006) [fRG & NRG]s
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Singular value decomposition
Diagonalize the tunneling matrix: Define new degrees of freedom The pseudo-spin is conserved in tunneling!
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Map to Anderson model Rotated magnetic field! z θ x
Generally Vup <> Vdown – ferromagnetic leads Generally h is not || to anisotropy axis x z θ Rotated magnetic field!
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Special case: an exact result
Degenerate levels: Spin is conserved → Friedel rule applies: SIC! Need a way to get magnetization M ! Local moment,
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Mapping to Kondo Hamiltonian
Schrieffer-Wolff for U >> Γ,h (local moment) Anisotropic exchange Effective field Anisotropy is RG irrelevant Silvestrov & Imry PRL (2000) Martinek et al PRL (2003)
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Main results An isotropic Kondo model in external field
Local moment here: An isotropic Kondo model in external field Use exact Bethe ansatz Key quantities Return back
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Main results: anisotropic Γ’s
Both competing scales depend on ε0 h ≈ TK => M=1/4 fRG h = 0
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Double dots: conclusions
Looking at the right angle makes old physics useful again Singular value decomposition (SVD) reduces dramatically the parameter space Accurate analytic expressions for linear conductance and occupations Future prospects: more levels add real spin non-equilibrium
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