1. 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

2. 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

3. 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

4. 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

5. Green’s functions Retarded Advanced Spectral function Zubarev (1960)
grand canonical

6. Equations of motion
Green function on the dot

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

8. 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)

9. Exact (but endless) hierarchy…

10. 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)

11. Where shall we stop?

12. 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)

13. 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

14. 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)

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

16. 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

17. 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 )

18. 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

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

20. 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!

21. 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

22. 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

23. 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

24. 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...

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

26. 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?

27. How to get rid of integration?P1 P2

28. 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:

29. 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

30. 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

31. 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

32. 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:

33. Results: occupation numbers
Compare to perturbation theory
Compare to Bethe ansatz
Gefen & Kőnig (2005)
Wiegmann & Tsvelik (1983)
Better than 3% accuracy!

34. 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

35. 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*

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

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

38. Results: susceptibility vs. TΓ
TK*

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

40. 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

41. Thanks!
Paper, poster & talk at kashcheyevs

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

43. Results: against Lacroix& MWL

44. Results: index=0 insufficiency

45. Temperature explained