1. Low energy approach for the SU(N) Kondo model
Christophe Mora, Xavier Leyronas, Nicolas Regnault
Laboratoire Pierre Aigrain, ENS, Paris
Many thanks to
Takis Kontos
ENS mesoscopic group
Aashish A Clerk, P. Vitushinsky
Karyn Le Hur

2. Outline of the talk Reminder: SU(2) Kondo model
Transport and Local Fermi liquid theory
SU(4) and SU(N)

3. SU(2) Kondo model

4. Coulomb blockade U (or EC) VSD R L VG Current flows for
Low energy sector
Charge excitation quenched: quantum dot acts as an impurity ½-spin.

5. Kondo hamiltonian
Charge excitation quenched: quantum dot acts as an impurity ½-spin.
Tunneling to leads: exchange interaction between conduction electrons and spin impurity.
Local interaction
Kondo antiferromagnetic coupling

6. Potential scattering
T-matrix
Pauli disappears…

7. Kondo scattering Kondo antiferromagnetic coupling
Fermi surface: IR divergences
Kondo (Prog. Theo. Phys., 1964)

8. Kondo screening Renormalization flow: J effectively large.
Susceptibility saturation indicates spin screening.
Ground state: many-body singlet.
Enhanced scattering and conductance (Kondo resonance).
L. Kouwenhoven and L. Glazman (2001)

9. Kondo in Quantum dots QD Carbon nanotube Scaling in T/TK
Goldhaber-Gordon et al (Nature, 1998)
Nygard, Cobden, Lindelof (Nature, 2000)
QD Carbon nanotube
L. Kouwenhoven and L. Glazman (2001)
Scaling in T/TK

10. Summary: Kondo effect U L R
Charge quenched, exchange with leads electrons.
Resonant Kondo scattering, formation of many-body singlet.
Conductance increases (in dots) at low T.

11. Transport and Local Fermi liquid theory
VSD VG

12. Local Fermi liquid picture
Nozières (1974)
Scattering via virtual polarization of singlet (energy TK).
imposed by Friedel sum rule.
For comparison: resonant level model.
But Kondo resonance is a many-body effect.

13. Floating of Kondo resonance
Where to put the Fermi energy (reference) ?
Out of equilibrium situation
Phase shift independent of Fermi level position L R ?

14. CFT and low energy theory
Conformal field theory provides description for the strong coupling fixed point.
free fermions with phase shift π/2 for SU(2) exactly screened case.
Affleck, Ludwig (1991)
Leading irrelevant operators guessed from CFT. Determine low energy properties.
Recovered with α1 = φ1
Integrable QFT technics: complete IR hamiltonian expansion extracted from Bethe ansatz.
Lesage, Saleur (PRL, 1999)

15. Local Fermi liquid picture
Nozières (1974)
Quasiparticle phase-shift:
Elastic
Inelastic L
odd part
even part R
, L/R leads separated:
, plane waves for scattering states

16. Fractional Shot Noise (T=0, V<<TK)
Sela, Oreg, von Oppen, Koch (PRL, 2007) cL
Linear transport: δ(0)=π/2 is sufficient.
perfect transmission and no noise.
Non-linear transport (~V3).
effective charge e* =S/ 2 IBS . e* =e for non-interacting electrons (only elastic scattering).
poissonian statistic for backscattering,
events with one/two electrons. cR L R eV
Gogolin, Komnik (PRL 2006) Golub (PRB 2006) Meir, Golub (PRL 2002)

17. Current operator L R Straightforward approach:
does not apply for T≠0 Noise (Nyquist Noise not recovered)
Kaminski, Nazarov, Glazman (PRB 2000)
Back to Büttiker but with even/odd channels.
all elastic scattering encoded in b

18. Current noise computation
Landauer-Büttiker recovered with
Interactions on top of that, full perturbative Keldysh calculation is required.

19. Fano factor Effective charge Important temperature corrections
Mora, Leyronas, Regnault (PRL 2008)

20. SU(4) and SU(N) 1e

21. Orbital Kondo effect
Two degenerate subbands in carbon nanotubes (from K-K’).
Screening by leads requires orbital quantum number conservation during tunneling.
Jarillo-Herrero et al (Nature 2005)
Electron exchange couples all states
Much larger Kondo temperature 1e

22. Orbital +spin: SU(4) Kondo ?
Choi, Lopez, Aguado (PRL 2005)
Orbital degeneracy, enhanced Kondo temperature
Experimental observations
Four peaks splitting in magnetic field
Jarillo-Herrero et al (Nature 2005)
Sasaki, Amaha, Asakawa, Eto, Tarucha (PRL, 2004)
Makarovski, Zhukov, Liu, Finkelstein (PRB, 2007)
SU(4) symmetry difficult to prove.
Strong competition with two-level SU(2),
SU(4) unstable (asymmetry or non-orb-sel-tun)
Lim et al (PRB 2006)

23. SU(4) at low energy DOS Friedel sum rule implies δ(0)=π/4.
Lim et al (PRB 2006)
SU(4) at low energy
DOS
Friedel sum rule implies δ(0)=π/4.
Transmission T0 =1/2
Partition noise at strong coupling, S0=2 e3 V/h, does not vanish like SU(2).
Non-linear transport, δI, δS ~V3
Effective Kondo resonance above Fermi level.
model is not p-h symmetric.
FL description: lowest order.
Not sufficient for FL corrections !

24. FL corrections: second generation
elastic
inelastic
Kondo resonance ‘floating’:
Current approach (CFT): rd Casimir

25. Conductance for SU(N) SU(2) case SU(4) case SU(N)
p electrons on the dot

26. Universal ratio for SU(N)
Ratio obtained by comparison with Bethe-Ansatz solution (energy as a function of magnetic field).
Bazhanov, Lukyanov, Tsvelik (PRB 2003)
at large N
Comparison with large N approach. Calculation of Lorentz ratio. Both coincide !
Houghton, Read, Won (PRB 1987)

27. Universal ratio for SU(4)
Mora, Leyronas, Regnault (PRL 2008)
Vitushinsky, Clerk, Le Hur (PRL 2008)

28. Noise measurements
Noise measurements performed at LPA.
16/19

29. Noise measurements (II)
Conductance
Noise
T=1.4K
TK=3.45K
T=1.4K
TK=3K

30. Conclusion
General framework for low energy properties of SU(N) screening.
Effective Fano factor characterizes non-linear transport, effective charges can be extracted.
Important temperature corrections to shot noise.
Vitushinsky, Clerk, Le Hur (PRL 2008)
Mora, Leyronas, Regnault (PRL 2008)

32. L. Kouwenhoven and L. Glazman (2001)
Felsch (Z. Phys., 1978)
Winzer (Z. Phys., 1973)
D. Goldhaber-Gordon et al (Nature, 1998)
Nygard, Cobden, Lindelof (Nature, 2000)
Sasaki, Amaha, Asakawa, Eto, Tarucha (PRL, 2004)
Zarchin, Zaffalon, Heiblum, Mahalu, Umansky (PRB, 2008)
Makarovski, Zhukov, Liu, Finkelstein (PRB, 2007)

33. Metallic Alloys
Measurements of susceptibility in Rare-earth (La, Ce)B6
Felsch (Z. Phys., 1978)
Deviations from Curie-Weiss law
Measurements in (La, Ce)Al2
Minimum in resistivity
Log behavior
Winzer (Z. Phys., 1973)