Equations-of-motion technique applied to quantum dot models

Slava Kashcheyevs

Amnon Aharony

Ora Entin-Wohlman

cond-mat/0511656

Phys. Rev. B 73 (2006)

Thursday seminar at

March 9, 2006

Paradigm: the Anderson model

Anderson (1961) – dilute magnetic alloysGlazman&Raikh, Ng&Lee (1988) – quantum dots

• Generalizations– Structured leads (“mesoscopic network”)– Multilevel dots / multiple dots with capacitative / tunneling interactions – Spin-orbit interactions

• Essential: many-body interactions restricted to the dot

How to treat Anderson model?• Perturbation theory (PT) in Γ, in U

analyticcontrollabletricky to extend into strong coupling (Kondo) regime

• Map to a spin model and do scaling

• Numerical Renormalization Group (NRG)accurate low energy physicsinherently numerical

• Bethe ansatzexact analytic solutionintegrability condition too restrictive, finite T laborious

• Equations of motion (EOM)analyticas good as PT when PT is valid

? not controlled for Kondo, but can give reasonable answers

Outline

• Get equations– Definitions and the exact EOM hierarchy– Truncation: self-consistent vs. perturbative

• Solve equations– Analysis of sum rules => bad news– Exact solution for => some good news

• Kondo physics with EOM– pro and con

Green’s functions• Retarded

• Advanced

• Spectral function

grand canonical

Zubarev (1960)

Equations of motion

Green function on the dot

Green function in simple cases

• No interactions (U=0)

D

Γ fully characterizes the leads

Wide-band limit: – approximate

Green function in simple cases

• Small U – approximate the extra term

Strict 1st order Self-consistent (Hartree)

Anderson (1961)

Decouple via Wick

Expand to 1st order

Exact (but endless) hierarchy

…

•

General term in the EOM

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)

• increases the total number of operators by adding two extra d’s

• Not more than can accumulate on the lhs of GF (finite Hilbert space on the dot!)

• Vk transforms d’s into lead c’s that do accumulate

Where shall we stop?

Decoupling• Stop before we get 6 operator functions

“D.C.Mattis scheme”:Theumann (1969)

spin conservation

• Use values

Meir, Wigreen, Lee(1991)

Meir-Wingreen –Lee (1991)

• Well characterizes Coulomb blockade downs to T ~ Γ • Popular and easy to use• Would be exact to , if one treated to 1st order

• Often referred to as “…works quantitatively at T > TK, and qualitatively at T< TK” – a misleading statement

Decoupling• Stop before we get 6 operator functions

“D.C.Mattis scheme”:Theumann (1969)

spin conservation

• Use values

Meir, Wigreen, Lee(1991)

• Demand full self-consistency

Appelbaum&Penn (1969); Lacroix(1981)Entin,Aharony,Meir (2005)

Self-consistent equations

Self-consistent functions:

Level positionZeeman splitting The only input parameters

Outline

• Get equations– Definitions and the exact EOM hierarchy– Truncation: self-consistent vs. perturbative

• Solve equations– Analysis of sum rules => bad news– Exact solution for => some good news

• Kondo physics with EOM– pro and con

• Relations at T=0 and Fermi energy:

• MWL approach gives .

Will self-consistency improve this?

First test: the sum rules

(For simplicity, look at the wide band limit )

0 “Unitarity” condition

Friedel sum rule

Langreth (1966)

Exploit low T singularities

Integration with the Fermi function:

T=0

P and Q develop logarithmic singularities at T=0 as

when either of these is 0, will have an equation for

Results for the sum rules• Expected:

• For and

• For and

“Unitarity” is OK

“Unitarity” is OK

Friedel implies:

Field-independent magnetization !

Particle-hole symmetry

• This implies symmetric DOS:

middle of Coulomb blockade valley

no Zeeman splitting

symmetric band

Exact cancellation in numerator & denominator separately at any T!

Particle-hole symmetry

Temperature-independent (!) Green function

At T=0 the “unitarity” rule is broken:

• The problem is mentioned in Dworin (1967), Appelbaum&Penn (1969), but in no paper after 1970!

• The Green function of Meir, Wingreen & Lee (1991) gives the same

Sum rules: summary

“Unitarity”

Friedel ?

“Unitarity” Friedel (“softly”)

T

“Unitarity”Friedel

In this plane, and limits do not commute

T=0 plane

Ouline

• Get equations– Definitions and the exact EOM hierarchy– Truncation: self-consistent vs. perturbative

• Solve equations– Analysis of sum rules => bad news– Exact solution for => some good news

• Kondo physics with EOM– pro and con

Exactly solvable limit

• Requires and wide-band limit

• Explicit quadrature expression for the Green function

• Self-consistency equation for 3 numbers (occupation numbers and a parameter)

• Will show how to …. remove integration

remove non-linearitySkip to Results...

Retarded couples to advanced and vice versa

Infinite U limit + wide band

A known function:

How to get rid of integration?

• Can we write the equations as algebraic relations between functions defined on the upper and lower edges of the cut?

Does not work for the unknown function:

How to get rid of integration?P1

P2

How to get rid of integration?• Introduce two new unknown functions

Φ1 and Φ2 (linear combinations of P and I),

• and two known X1 and X2 such that:

• The function must be a polynomial!

• Considering gives ,where r0 and r1 are certain integrals of the unknown Green function

Cancellation of non-linearityClear fractions and add:

Riemann-Hilbert problem• Remain with 2 decoupled linear problems:

A polynomial! From asymptotics,

Explicit solution!

• Expanding for large z gives a set of equations for a1, r0, r1 and <nd>

• The retarded Green function is given by

Outline

• Get equations– Definitions and the exact EOM hierarchy– Truncation: self-consistent vs. perturbative

• Solve equations– Analysis of sum rules => bad news– Exact solution for => some good news

• Kondo physics with EOM– pro and con

Results: density of states

• Zero temperature• Zero magnetic field• & wide band

Level renormalization

Changing Ed/Γ

Looking at DOS:

Ed / ΓEnergy ω/Γ

Fermi

Results: occupation numbers

• Compare to perturbation theory

• Compare to Bethe ansatz

Gefen & Kőnig (2005)

Wiegmann & Tsvelik (1983)

Better than 3% accuracy!

Results: Friedel sum rule

• “Unitarity” sum rule is fulfilled exactly:

• Use Friedel sum rule to calculate

Good – for nearly empty dot

Broken – in the Kondo valley

Results: Kondo peak meltingAt 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*

Magnetic susceptibility

• Defined as

• Explicit formula obtained by differentiating equations for with respect to h.

Wide-band limit χ

Results: magnetic susceptibility

!

Bethe susceptibility in the Kondo regime ~ 1/TK

Our χ is smaller, but on the other hand TK* <<TK ?!

Results: susceptibility vs. T

Γ

TK*

Results: MWL susceptibility

MWL gives non-monotonic and even negative χ for T < Γ

Conclusions!

• EOM is a systematic method to derive analytic expressions for GF

• Wise (sometimes) extrapolation of perturbation theory

• Applied to Anderson model,– excellent for not-too-strong correlations– fair qualitative picture of the Kondo regime– self-consistency improves a lot

Paper, poster & talk at kashcheyevs

Results: DOS ~ 1/log(T/TK*)2

Results: against Lacroix& MWL

Results: index=0 insufficiency

Temperature explained