Vyacheslavs (Slava) Kashcheyevs

December 18th, 2006

Collaboration:Amnon Aharony (BGU+TAU)Ora Entin-Wohlman (BGU+TAU)Avraham Schiller (Hebrew Univ.)

Correlations in quantum dots:How far can analytics go?

Outline• Physics of the Anderson model

– 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

Quantum dots

QD

GaAs AlGaAs

Gates

Lead

LeadElectron gas plane

• Tune: gate potentials, temperature, field…

• Measure: I-V curves, conductance G…

• Aharonov-Bohm interferometry, dephasing, coherent state manipulation…

Building models

• Quantum dots– Mesoscopic:

many levels involved, statistical description– Microscopic: few levels, individual properties

• Tunneling Hamiltonian approach

Anderson model: the leads

• A set of energy levels

LeadLead

μ

Anderson model: the dot

LeadLead

μ

QD

• An energy level ε0

Anderson model: the dot

LeadLead

μ

QD

• An energy level ε0

ε0

1

2

n

μ

f (ε0)

T

Anderson model: the dot

LeadLead

μ

QD

• An energy level ε0

ε0

1

2

n

μ

f (ε0)

T

Anderson model: the dot

LeadLead

μ

QD

• An energy level ε0

ε0

1

2

n

μ

f (ε0)

T

Interactions!

U

μ

USpin-charge separation or “Mott transition”

Anderson model: tunneling

LeadLead QD

• Tunneling

ε0

1

2

n

μ μ

UVL VR

Tunneling rate Γ ~ ρ|V|2

Γ

Γ

Quantum fluctuations:– of charge Γ > T – of spin U > Γ > T

Anderson model: spin exchange

LeadLead QD

• Fix the spin on the dot

• Opposite spin in the leads can lower energy!

ε0

1

2

n

μ μ

U

Anderson model: spin exchange

LeadLead QD

• Fix the spin on the dot

• Opposite spin in the leads can lower energy!

ε0

1

2

n

μ μ

U

Virtual transition:

Anderson model: spin exchange

LeadLead QD

• Fix the spin on the dot

• Same spin can’t go!

ε0

1

2

n

μ μ

U

Virtual transition:

Effective Hamiltonian: Kondo

LeadLead QD

• Ferromagnetic exchange interaction!

Effective Hamiltonian: Kondo

LeadLead QD

• Ferromagnetic exchange interaction!

← can fix S with h >> J

h

1/2

<S>

J

½ – O(J)

?

0

The Kondo effect

LeadLead QD

• |↑↓> and |↓↑> are degenerate (Sz=0 of S=0,1)

• except for virtual excitations ~ J !

h

1/2

<S>

J

½ – O(J)

?

0 TK

singlet-triplet splitting

~ h/TK

Outline• Physics of the Anderson model

– 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

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 functionsexact for U; a low order (Hartree mean field) gives local momentas good as PT when PT is valid

? can we get Kondo at higher orders?

• Renormalization Groupperturbative (in Γ) RG => Kondo Hamiltonian + PM scalingperturbative (in U) RG => functional RG (semi-analytic)Wilson’s Numerical RG – high accuracy, but numerics only

• Bethe ansatz: exploits integrabilityexact (!) solution, many analytic resultsintegrability condition too restrictive, finite very T laborious

Outline• Physics of the Anderson model

– 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 Green functions• Retarded

• Advanced

• Spectral function

grand canonical

Zubarev (1960)

step function

Dot’s GF

• Spectral function → Density of states

• Occupation number → Local charge & spin

• Friedel-Langreth sum rule (T=0)

conductance at T=0is proportional ~ ρ(μ)!

Equations of motion

• Example: 1st equation for

Full solution for U=0

bandwidth D

Γ

Lead self-energy function

Lorenzian DOS Large U should bring

ε0 +Uε0 ω=0Fermi

hole excitations

electron excitations

spin-flip excitations

Full hierarchy

…

A general term

Contributes to a least m-th order in Vk and (n+m–1)/2-th order in U !

m = 0,1, 2… lead operators n = 0 – 3 dot operators

Dworin (1967)

Certain order of EOM truncation ↔ certain order of perturbation theory!

Decoupling

• Use values

Meir, Wigreen, Lee (1991)

Linear = easy to solve Fails at low T – no Kondo

Decoupling• Use mean-field for at most 1 dot operator:

“D.C.Mattis scheme”:Theumann (1969)

• Demand full self-consistency

Significant improvement Hard-to-solve non-linear integral eqs.

The self-consistent equations

Self-consistent functions:

Level positionZeeman splitting The only input parameters

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

Outline• Physics of the Anderson model

– 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

EOMs: Results

• Zero temperature• Zero magnetic field• & wide band

Level renormalization

Changing Ed/Γ

Looking at DOS:

Ed / ΓEnergy ω/Γ

Fermi

odd

even

Results: occupation numbers

• Compare to perturbation theory

• Compare to Bethe ansatz

Gefen & Kőnig (2005)

Wiegmann & Tsvelik (1983)

Better than 3% accuracy!

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

Results: melting of the peakAt small T andnear Fermi energy, parameters in the solution combine as

Smaller than the true Kondo T:

2e2/h conduct.

~ 1/log2(T/TK)

DOS at the Fermi energy scales with T/TK*Experiment: van der Wiel et al., Science 289, 2105 (2000)

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

Outline• Physics of the Anderson model

– 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

Double dots: a minimal model• Two orbital levels• Two leads• Inter-dot

– repulsion U– tunneling b

• Aharonov-Bohm flux

(wide band)

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

Singular value decomposition

• Diagonalize the tunneling matrix:

• Define new degrees of freedom

• The pseudo-spin is conserved in tunneling!

Map to Anderson model

x

z

θ

Rotated magnetic field!

Special case: an exact result• Degenerate levels:

• Spin is conserved → Friedel rule applies:

Need a way to get magnetization M !

Local moment,

Mapping to Kondo Hamiltonian• Schrieffer-Wolff for U >> Γ,h (local moment)

Silvestrov & Imry PRL (2000)

Martinek et al PRL (2003)

• Anisotropic exchange• Effective field• Anisotropy is RG irrelevant

Main results

• An isotropic Kondo model in external field

• Use exact Bethe ansatz

• Key quantities

• Return back

Local moment here:

Main results: anisotropic Γ’s• Both competing scales depend on ε0

fRG

h ≈ TK => M=1/4

h = 0

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