1. The Two Channel Kondo Effect (The breakdown of the Fermi liquid paradigm in quantum dots: theory and experiment) Department of Condensed Matter Physics Weizmann Institute of Science HRI, 8 Feb 2008

2. Outline Ancient and modern history of the Kondo effect Shot noise of 5/3 in the Kondo limit (theory and experiments) The Fermi liquid and the Non Fermi liquid theories The two-channel Kondo (2CK) theoretical predictions 90 136602 (2003). Experimental observation of the two channel Kondo state 446, 167 – 171 (2007). Summary

3. Resistance of metals at low temp Temperature (K) 10 4 R(T) R(273) 1 23 4 5 26.4 26.5 De Hass, de Bour, de Berg (1934)

4. M. Sarachik et al. 1964 Iron ’ s Concen. in Mo.8 Nb.2 Fe is an example for a 3d transition metals

5. The s-d model FERMI SEA (Conduction band) Local spin of conduction electrons Impurity spin J σS, J>0

6. Kondo Solution Band Width Density of states

7. The Kondo Problem Perturbation theory fails at the Kondo temp. J 2 = J 3 Log(D/T)  T K =D e -1/ J What happens below T K ?

8. Developments Scaling and Renormalization Group Exactly solvable points Coulomb Gases Numerical Renormalization Group Bosonization (Schotte and Schotte) Bethe ansatz solution (Andrei, Wiegman- Tsvelik )

9. Kondo Resonance Conduction spin Impurity spin J σS, J>0

10. A universal bound state is formed between the local spin and the conduction electrons Density of states Energy EFEF

11. Fermi-Edge Singularity (FES) FERMI SEA (Conduction band) EFEF EgEg Attraction between valance hole and conduction electrons. Orthogonality “catastrophe” Sudden potential + Many body effects  E F +E g Absorption

12. Single Channel Kondo and Many FES Time J σS= J z σ z S z + J - σ + S - + J + σ - S + Density of states Energy EFEF

13. Origin of Spin-Spin interaction The Anderson Model  s-d model Electron processes JzσzSzJzσzSz J-σ+S-J-σ+S-

14. Origin of Spin-Spin interaction The Anderson Model  s-d model Electron processes Hole processes t- coupling  d - impurity level U-charging energy dd U+2  d t

15. Theory: a small quantum dot forms the local impurity state [Ng & Lee, Glazman & Raikh (1988)] Experiments 1CK in a quantum dot

16. Strongly coupled and small dots 1 μm Exp:Yang Ji, M. Heiblum et al. Sci. 290, 779. T K =D e -1/ J J~t 2 /U

17. 1998

18. 2000

19. In quantum dots we can tune and control many parameters

20. Phase measurements Gerland, Costi, von Delft & Oreg PRL 84, 3710 2000 Exp:Yang Ji, M. Heiblum et al. Sci. 290, 779 2000 0 µV; -4 4 0 µV; -80 µV; -1 1 00 µV Exp: Zafalon, et al. 2007

21. 2004

22. Fractional Noise in the Kondo Limit Eran Sela, YO, von Oppen, Koch PRL 2006

23. Nozières FL Hamiltonian αε πνT K ε ε ε ε βnσβnσ πν 2 T K =(1-e 2iδσ )/(2πiν)=-δ σ /πν δσ= δσ= αε TKTK βnσβnσ νTKνTK Elastic part

24. Floating of the Kondo resonance δn δn δε δ σ (ε=0, n=0) = δ σ (δε, n=νδε) δσ= δσ= αεαε TKTK βnσβnσ νTKνTK α=βα=β Keldysh vs FGR

25. Relation of ψ particles to L-R movers Scattering from R  L causes backscattering current Both α and β contribute to the backscattering of R  L In the unitary limit: no scattering between L and R  ψ is free S D Right movers Left movers For symmetric coupling ψ =(L+R)/√2 Long rigorous derivation …

26. α β2β2 β1β1 β2β2 α t α β1β1 β2β2

27. Symmetry breaking effects 1.Left-Right symmetry of the dot 2.SU(2) symmetry (magnetic field) 3.Particle-hole symmetry δθδhδrδθδhδr

28. Nature Physics 2007

29. Fermi Liquid vs. Non Fermi Liquid

30. Fermi Liquid (FL) Concept of quasi particles. 1/(Life time) << Typical energy Fermi Liquid vs. Non Fermi Liquid p F p N(p) 1 S D dI/dV sd Ө= Max[T,V sd ] dI/dV sd Ө=Max[T, V sd ] Non FL Superconductors. Luttinger liquid in 1D. FQHE liquids. Low D systems + interaction + disorder. Multi-channel Kondo. const- Ө 2 const- Ө 1/2

31. J 31 YO and David Goldhaber-Gordon PRL 2003 Lead-1 Lead-2 Lead-3 J 22 J 31 J 13 =J 11 J 33 J 31 J 13 ~V 3 V 1 V 1 V 3 =V 1 V 1 V 3 V 3 ~J 11 J 33 V3V3 V1V1 ECEC E C >T

32. The two channel Kondo Hamiltonian H=Σ ε k i † kα i kα + Σ ε k f † kα f kα J i σ i S+J f σ f S

33. Multi-Channel Kondo  RG: (Nozieres – Blandin, Zawadowski). NRG.  Bethe Ansatz (Andrei, Wiegmann-Tsvelik)  Conformal Field Theory (Affleck - Ludwig)  “Pseudo Particle” or “Slave Bosons” (Read - Coleman, Ruckenstein - Cox)  Bosonization (Kivelson - Emery) Spin flavor Majorana Fermion

34. 2CK fixed point and a local Majorana Total charge ρ i ↑ + ρ i ↓ + ρ f ↑ +ρ f ↓ Total spin ρ i ↑ - ρ i ↓ + ρ f ↑ -ρ f ↓ Flavor ρ i ↑ + ρ i ↓ - ρ f ↑ -ρ f ↓ Spin flavor ρ i ↑ - ρ i ↓ - ρ f ↑ +ρ f ↓ Bosonization  a series of rotations and translations  Refermionization iΣΨ † sf ∂Ψ sf + J + [Ψ † sf (0) + Ψ sf (0)][S + -S - ] η + ia - iη - a + η=η † {η,η † }=1

35. 2-Channel vs. 1-Channel Kondo Specific Heat Conductance Spin Susceptibility Entropy Two ChannelSingle Channel

36. 2CK 1CK J finite lead J infinite lead dI/dV Θ J=JJ=J J>JJ>J J<JJ<J

37. Observation of the two channel Kondo effect R. M. Potok, I. G. Rau, Hadas Shtrikman, Yuval Oreg and D. Goldhaber-Gordon Nature 446, 167 – 171 (2007). S D 1μm1μm c

38. g(e 2 /h)

39. dI/dV Θ J>JJ>J J<JJ<J g(0,T)-g(V sd,T) T α

40. J≈JJ≈J Scaling exponent and Scaling function A new metallic state of matter was observed

42.  What is the effect of magnetic field ( Kikoin and YO PRB submitted )?  What is the shot noise ( 1CK Sela, YO, et al. PRL 2006 ), 2CK?  What happens when the temperature is smaller than the level spacing in the finite reservoir.  Can we have more channels?  Can we design novel many body states leading to a new comprehension of strongly correlated systems? Summary and F.A.Q.

43. Magnetic field induced two-channel Kondo effect in multiple quantum dots Kikoin & YO

44. T K2 vs. T K1 T K2  E c

45. A simple realization of MCK YO and David Goldhaber-Gordon (PRL 2003) d N-1 N 2 1 Pistil: small dot Petal: large dots or leads l

46. Due to correlations induced by the screening cloud sometimes two electrons are scattered in pairs leading to an “effective charge of 5/3”. Ask Eran Sela.

47. Suggestions for 2CK Realizations Two level systems, “spins are isotropic channels” Coulomb blockade peak is a degenerate state Quadruple 2CK ( Cox ) Non equilibrium ( Wen ) Luttinger leads ( Kim )  Theo: Zawadowski, von Delft et al,  Exp: Dan Ralph and Burman.  The. Matveev et al. (Requires a large dot, and smooth contacts)  Exp. Devoret et al. Hard in real systems.  Schiller et al. Average level spacing Kondo temp. Charging energy Lead-Dot Coupling const ~100 Publications/attempts to realize MCK