Slava Kashcheyevs Avraham Schiller Amnon Aharony Ora Entin-Wohlman

1. Interference and correlations in two-level dots Slava Kashcheyevs Avraham Schiller Amnon AharonyOra Entin-Wohlman Phys. Rev. B 75, 115313 (2007) Also: Silvestrov & Imry, PRB 75, 115335 (2007) Lee & Kim, PRL 98, 186805 (2007)

2. Conductance gate voltage Phase Motivation “Phase lapse” Avinun-Kalish et al.,Nature 436 (2005)Schuster et al., Nature 385 (1997)

3. Destructive interference – several paths through the dot Non-interacting model gives either 0 or πphase change between the resonances ε1 U ε2 Motivation continued Explicit on-siteCoulomb interaction Entin-Wohlman, Hartzstein & Imry (1986)Silva, Oreg & Gefen (2002)Entin-Wohlman,Aharony,Levinson&Imry (2002) Interaction-based qualitative explanation of the phase lapse universality: Silvestrov & Imry PRL 85 (2000)

4. ε1 ε2 Motivation continued • Non-monotonic level fillingand population inversion • Silvestrov & Imry (2000) [mechanism & PT] • König & Gefen PRB 71 (2005)[perturbation in tunneling] • Sindel, Silva, Oreg & von Delft PRB 72 (2005) [NRG & Hartree-Fock] • Transmission zeros and “phase lapses” • Silvestrov & Imry (2000) • Meden & Marquardt PRL (2006)[functional RG and NRG] • Golosov & Gefen PRB 74(2006)[Hartree-Fock (mean field)] • Karrasch,Hecht,Weichselbaum,Oreg, vonDelft & Meden PRL(2007) [NRG & fRG] • Orbital Kondo physics (“Correlation-induced” resonances) U • Two orbital levels • Two leads • On-site repulsion U • Spinless electrons

5. Questions to answer • Non-monotonic level fillingand population inversion • Silvestrov & Imry (2000) [mechanism & PT] • König & Gefen PRB 71 (2005)[perturbation in tunneling] • Sindel, Silva, Oreg & von Delft PRB 72 (2005) [NRG & Hartree-Fock] • Transmission zeros and “phase lapses” • Silvestrov & Imry (2000) • Meden & Marquardt PRL (2006)[functional RG and NRG] • Golosov & Gefen PRB 74(2006)[Hartree-Fock (mean field)] • Karrasch,Hecht,Weichselbaum,Oreg, vonDelft & Meden PRL(2007) [NRG & fRG] • Orbital Kondo physics (“Correlation-induced” resonances) • Accurate methods… • either numrical only • or too narrow validity range • Hard to sample parameter space • symmetric (1-2 or L-R) cases are non-generic • Underlying energy scales • Role of many-body correlations • Unifying geometrical picture

6. Outline Original spinless 2 levels x 2 leads Observablesn1, n2, t Exact mapping Inverse mapping, Friedel sum rule Equivalent Anderson model 1 spinful level x 1 ferromagnetic lead V↑ = V↓ Use exact solution(Bethe ansatz) Schrieffer-Wolff transformation U >> Γ Anisotropic Kondo model in a titled magneticfield Isotropic Kondo with a field

7. ε0+h/2 U ε0–h/2 The model: notation • Two orbital levels • Two leads • Level spacing h • On-site Coulomb U • No symmetry imposed on aαi (wide band, D>>U)

8. Singular value decomposition • Diagonalize the tunneling matrix: • Define new degrees of freedom • The pseudo-spin is conserved in tunneling!

9. Singular value decomposition • Diagonalize the tunneling matrix: • Define new degrees of freedom • Rd, Rl are orthogonal matrices

10. scalar spin vector in a tilted magnetic field Map onto Anderson two preferred directions!

11. Outline Original spinless 2 levels x 2 leads Observablesn1, n2, t Exact mapping Inverse mapping, Friedel sum rule Equivalent Anderson model 1 spinful level x 1 ferromagnetic lead V↑ = V↓ Use exact solution(Bethe ansatz)

12. “Standard” Anderson: In terms of original couplings: At T=0, an exact solution is possible for n1, n2 Numerical solution of Bethe ansatz equations fixed Solvable case: isotropic V one preferred direction Wiegman (1980); Okiji & Kawakami (1982)

13. Local moment  single occupancy Polarization  charge localization Correlation-driven competition (see later) No phase lapse Γ=πρ|V|2 Γ U Exact results for isotropic AM n1n2 n1+n2 ≈ 1 |t|2 arg t Friedel-Langrethsum rule: Glazman & Raikh

14. Outline Original spinless 2 levels x 2 leads Observablesn1, n2, t Exact mapping Inverse mapping, Friedel sum rule Equivalent Anderson model 1 spinful level x 1 ferromagnetic lead V↑ = V↓ Exact solution(Bethe ansatz) Schrieffer-Wolff transformation U >> Γ Anisotropic Kondo model in a titled magneticfield Isotropic Kondo in with a field

15. Magnetic insights… • A quantum dot with ferromagnetic leads • V↑ ≠ V↓ generates additional local field • the physics: renormalization of level positions • We shall translate back to the charge problem: • Polarization in magnetic field competes with Kondo screening • 2D twist: the bare & the extra fields are not aligned => spin rotations effective Zeeman field Martinek et al., PRL91127203; 247202 (2003) Pasupathy et al., Science306, 86 (2004)

16. Mapping onto a Kondo model • Schrieffer-Wolff in CB valley (U >>Γ, h) … • anisotropic exchange • effective field

17. anisotropic exchange • effective field Mapping onto a Kondo model • Schrieffer-Wolff in CB valley (U >>Γ, h) … • Poor man’s scaling gives TK • Anisotropy is RG irrelevant • use results for isotropic Kondo model in

18. generalized Glazman-Raikh phase shifts via sum rule Geometrical interpretation • Magnetization is determined by the field • Known function MK • Project onto original1-2 direction Bethe ansatz for isotropic Kondo modelby Andrei &Lowenstein (1980) Transmission L-R:

19. 0.47 0.25 U/Γtot =3 0.08 0.16 An example Numbers from Fig.5 of PRL 96, 146801 (2006) θd=31º θl = 62º SVD angles reflect asymmetry in tunneling Γ↑ = 0.97 Γtot Γ↓ = 0.03 Γtot Changing gate voltage ε0 leads to effective field rotation!

20. Small spacing : correlations h=0.01

21. Population inversionSilvestrov & Imry (2000) “Correlation-induced resonances”Meden & Marquardt (2006) h=0.01  Phase lapse by πSilvestrov & Imry (2000) Small spacing : correlations htot TK θh M n1-n2 |t|2 h=0.01 ε0= – U/2 ε0

22. Göres et al., PRB 62, 2188(2000) θl θd+90º Intermediate spacing: rotations htot θh M Fano resonances! n1-n2 |t|2 h=0.1 ε0= – U/2 ε0

23. Relevant energy scales • Range of ε0-dependent component • Transversal projection of level spacing • Kondo correlation scale • Occupations numbers and transmission amplitudeare always* smooth • Generic, sharp π-jump of phase for • The population inversion and the phase lapse need not to coincide

24. heff≈TK => M=1/4 fRG heff >TK heff >TK heff= 0 Compare to other methods • Both heffand TKdepend on ε0 but h = 0

25. Summary and outlook • Results • Unified picture of both correlated and perturbative behavior • Accurate analytical estimates • Work in progress • many levels & statistics of phase lapses • Other issues • charge fluctuations (mixed valence)? • physical spin?

26. Thanks! Kashcheyevs

27. Only one combination couples to the dot Scattering of the coupled mode Langreth (1966) For , “unitarity limit” L R VR VL Glazman-Raikh as 2x1 SVD Glazman-Raikh rotation (1988)

28. Example: h=0 (degenerate) htot TK θh M n1-n2 |t|2 ε0= – U/2 ε0

29. ↑-↓ phase shift difference Conductance in isotropic case • For h || z, spin is conserved • Rotations imply • Friedel sum rule π/2 0

30. here: Local moment Bethe results • An isotropic Kondo model in external field • Use exact Bethe ansatz • Key quantities • Return back