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Equations-of-motion technique applied to quantum dot models Slava Kashcheyevs Amnon Aharony Ora Entin-Wohlman cond-mat/ Phys. Rev. B 73 (2006) Thursday seminar at March 9, 2006

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Paradigm: the Anderson model Anderson (1961) – dilute magnetic alloys Glazman&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

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How to treat Anderson model? Perturbation theory (PT) in Γ, in U analytic controllable tricky to extend into strong coupling (Kondo) regime Map to a spin model and do scaling Numerical Renormalization Group (NRG) accurate low energy physics inherently numerical Bethe ansatz exact analytic solution integrability condition too restrictive, finite T laborious Equations of motion (EOM) analytic as good as PT when PT is valid ?not controlled for Kondo, but can give reasonable answers

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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

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Greens functions Retarded Advanced Spectral function grand canonical Zubarev (1960)

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Equations of motion Green function on the dot

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Green function in simple cases No interactions (U=0) D Γ fully characterizes the leads Wide-band limit: – approximate

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Green function in simple cases Small U – approximate the extra term Strict 1 st orderSelf-consistent (Hartree) Anderson (1961) Decouple via Wick Expand to 1 st order

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Exact (but endless) hierarchy …

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General term in the EOM Contributes to a least m- th order in V k 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!) V k transforms d s into lead c s that do accumulate

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Where shall we stop?

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Decoupling Stop before we get 6 operator functions D.C.Mattis scheme: Theumann (1969) spin conservation Use values Meir, Wigreen, Lee(1991)

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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 1 st order Often referred to as …works quantitatively at T > T K, and qualitatively at T< T K – a misleading statement

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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)

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Self-consistent equations Self-consistent functions: Level position Zeeman splitting The only input parameters

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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

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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)

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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

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Results for the sum rules Expected: For and Unitarity is OK Friedel implies: Field-independent magnetization !

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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!

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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

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Sum rules: summary Unitarity Friedel ? Unitarity Friedel (softly) T Unitarity Friedel In this plane, and limits do not commute T=0 plane

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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

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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-linearity Skip to Results...

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Retarded couples to advanced and vice versa Infinite U limit + wide band A known function:

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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:

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How to get rid of integration? P1P1 P2P2

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Introduce two new unknown functions Φ 1 and Φ 2 (linear combinations of P and I), and two known X 1 and X 2 such that:

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The function must be a polynomial! Considering gives,where r 0 and r 1 are certain integrals of the unknown Green function Cancellation of non-linearity Clear fractions and add:

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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 a 1, r 0, r 1 and The retarded Green function is given by

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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

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Results: density of states Zero temperature Zero magnetic field & wide band Level renormalization Changing E d /Γ Looking at DOS: E d / Γ Energy ω/Γ Fermi

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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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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

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Results: Kondo peak melting At small T and near Fermi energy, parameters in the solution combine as Smaller than the true Kondo T: 2e 2 /h conduct. ~ 1/log 2 (T/T K ) DOS at the Fermi energy scales with T/T K *

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Magnetic susceptibility Defined as Explicit formula obtained by differentiating equations for with respect to h. Wide-band limit χ

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Results: magnetic susceptibility ! Bethe susceptibility in the Kondo regime ~ 1/T K Our χ is smaller, but on the other hand T K * <

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Results: susceptibility vs. T Γ TK*TK*

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

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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

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Paper, poster & talk at kashcheyevs

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

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Results: against Lacroix& MWL

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Results: index=0 insufficiency

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Temperature explained

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