Department of Condensed Matter Physics Weizmann Institute of Science

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 104R(T) R(273) De Hass, de Bour, de Berg (1934) 26.5 26.4 4 2 3 5 1 Temperature (K)

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

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

6. Kondo Solution Band Width Density of states

7. The Kondo Problem Perturbation theory fails at the Kondo temp. J2=nJ3Log(D/T) TK=D e-1/nJ What happens belowTK?

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 σ•S, J>0

10. A universal bound state is formedbetween the local spin and the conduction electrons Density of states Energy EF

11. FERMI SEA (Conduction band) Absorption w EF+Eg • Attraction between valance hole and conduction electrons. • Orthogonality “catastrophe” Sudden potential + Many body effects Fermi-Edge Singularity (FES) EF Eg

12. J σ•S= JzσzSz+ J-σ+S-+ J+σ-S+ Time Energy EF Single Channel Kondo and Many FES Density of states

13. Origin of Spin-Spin interactionThe Anderson Model  s-d model JzσzSz J-σ+S- • Electron processes

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

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

16. Strongly coupled and small dots Exp:Yang Ji, M. Heiblum et al. Sci. 290, 779. TK=D e-1/nJ J~t2/U 1 μm

17. 1998

18. 2000

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

20. Exp: Zafalon, et al. 2007 Phase measurements Exp:Yang Ji, M. Heiblum et al. Sci. 290, 779 2000 Gerland, Costi, von Delft & Oreg PRL 84, 3710 2000 0 µV; -440 µV; -80 µV; -1100 µV

21. 2004

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

23. αε αε πνTK TK βnσ βnσ πν2TK νTK δσ= Nozières FL Hamiltonian Elastic part ε ε ε ε =(1-e 2iδσ)/(2πiν)=-δσ/πν

24. αε TK βnσ νTK δσ= Floating of the Kondo resonance δn δε δσ(ε=0, n=0)= δσ(δε, n=νδε) α=β Keldysh vs FGR

25. Relation of ψ particles to L-R movers Long rigorous derivation … Right movers S • 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 Leftmovers D For symmetric coupling ψ=(L+R)/√2

26. α β2 β1 α β2 α β1 β2 t

27. Symmetry breaking effects • Left-Right symmetry of the dot • SU(2) symmetry (magnetic field) • Particle-hole symmetry δθ δh δr

28. Nature Physics 2007

29. Fermi Liquid vs. Non Fermi Liquid

30. Fermi Liquid vs. Non Fermi Liquid N(p) 1 dI/dVsd p p dI/dVsd F S Ө=Max[T, Vsd] Ө= Max[T,Vsd] D • Multi-channel Kondo. const-Ө2 const-Ө1/2

31. V1 V3 YO and David Goldhaber-GordonPRL 2003 J31J13~V3V1V1V3=V1V1V3V3~J11J33 J31J13=J11J33 Lead-1 EC EC >T J31 Lead-3 J22 Lead-2

32. The two channel Kondo Hamiltonian H=Σεk i†kαikα+ Σ εkf†kαfkα Jiσi•S+Jfσf•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 chargeρi↑+ ρi↓+ρf↑+ρf↓ • Total spinρi↑- ρi↓ +ρf↑-ρf↓ • Flavor ρi↑+ ρi↓-ρf↑-ρf↓ • Spin flavorρi↑- ρi↓-ρf↑+ρf↓ Bosonization  a series of rotations and translations  Refermionization iΣΨ†sf ∂Ψ sf + J+[Ψ†sf(0) + Ψsf(0)][S+-S-] η+ ia- η=η† {η,η†}=1 iη- a+

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

36. 1CK J<J J=J J>J Jinfinite lead 2CK 1CK Jfinite lead dI/dV Θ

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

38. g(e2/h)

39. J<J dI/dV J>J Θ g(0,T)-g(Vsd,T) Tα

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

41. Dziękuje  감사합니다 どうもありがとう спасибо धन्यवाद Ic þancas do 謝謝

42. Summary and F.A.Q. • 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?

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

44. TK2 vs. TK1 TK2  Ec

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

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. Charging energy Lead-Dot Coupling const Kondo temp. Average level spacing Suggestions for 2CK Realizations ~100 Publications/attempts to realize MCK • 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.