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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**

Generalizations Structured leads (“mesoscopic network”) Multilevel dots / multiple dots with capacitative / tunneling interactions Spin-orbit interactions Essential: many-body interactions restricted to the dot Anderson (1961) – dilute magnetic alloys Glazman&Raikh, Ng&Lee (1988) – quantum dots

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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) as good as PT when PT is valid not controlled for Kondo, but can give reasonable answers

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**Outline Get equations Solve equations Kondo physics with EOM**

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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**Green’s functions Retarded Advanced Spectral function Zubarev (1960)**

grand canonical

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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 1st order Self-consistent (Hartree) Decouple via Wick Expand to 1st order Anderson (1961)

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**Exact (but endless) hierarchy**

…

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

General term in the EOM 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 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 ! Dworin (1967)

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Where shall we stop?

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**Decoupling Stop before we get 6 operator functions Use values**

“D.C.Mattis scheme”: Theumann (1969) Stop before we get 6 operator functions 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 1st order Often referred to as “…works quantitatively at T > TK, and qualitatively at T< TK” – a misleading statement

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**Demand full self-consistency**

Decoupling “D.C.Mattis scheme”: Theumann (1969) Stop before we get 6 operator functions 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**

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

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**Outline Get equations Solve equations Kondo physics with EOM**

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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**First test: the sum rules**

Langreth (1966) Relations at T=0 and Fermi energy: MWL approach gives Will self-consistency improve this? “Unitarity” condition Friedel sum rule (For simplicity, look at the wide band limit )

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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 “Unitarity” is OK Friedel implies: Field-independent magnetization !

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**Particle-hole symmetry**

middle of Coulomb blockade valley no Zeeman splitting symmetric band This implies symmetric DOS: 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 (“softly”) “Unitarity” Friedel**

T=0 plane “Unitarity” Friedel (“softly”) T “Unitarity” Friedel “Unitarity” Friedel ? In this plane, and limits do not commute

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**Ouline Get equations Solve equations Kondo physics with EOM**

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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**Infinite U limit + wide band**

Retarded couples to advanced and vice versa A known function:

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**How to get rid of integration?**

Does not work for the unknown function: Can we write the equations as algebraic relations between functions defined on the upper and lower edges of the cut?

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**How to get rid of integration?**

P1 P2

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**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:

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**Cancellation of non-linearity**

Clear fractions and add: The function must be a polynomial! Considering gives ,where r0 and r1 are certain integrals of the unknown Green function

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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 a1, r0, r1 and <nd> The retarded Green function is given by

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**Outline Get equations Solve equations Kondo physics with EOM**

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**

Ed / Γ Energy ω/Γ Fermi Zero temperature Zero magnetic field & wide band Level renormalization Changing Ed/Γ Looking at DOS:

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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**

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*

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**Magnetic susceptibility**

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

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**Results: magnetic susceptibility**

! Bethe susceptibility in the Kondo regime ~ 1/TK Our χ is smaller, but on the other hand TK* <<TK ?!

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**Results: susceptibility vs. T**

Γ 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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Thanks! Paper, poster & talk at kashcheyevs

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