Slava Kashcheyevs Avraham Schiller Amnon Aharony Ora Entin-Wohlman Interference and correlations in two-level dots Phys. Rev. B 75, (2007) Also: Silvestrov & Imry, PRB 75, (2007) Lee & Kim, PRL 98, (2007) gate voltage Conductance Motivation Avinun-Kalish et al.,Nature 436 (2005) Schuster et al., Nature 385 (1997) Phase Phase lapse Motivation continued Entin-Wohlman, Hartzstein & Imry (1986) Silva, Oreg & Gefen (2002) Entin-Wohlman,Aharony, Levinson&Imry (2002) Destructive interference several paths through the dot Non-interacting model gives either 0 or phase change between the resonances U Explicit on-site Coulomb interaction Interaction-based qualitative explanation of the phase lapse universality: Silvestrov & Imry PRL 85 (2000) 11 22 Motivation continued U Non-monotonic level filling and population inversion Silvestrov & Imry (2000) [mechanism & PT] Knig & 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) 11 22 Two orbital levels Two leads On-site repulsion U Spinless electrons Non-monotonic level filling and population inversion Silvestrov & Imry (2000) [mechanism & PT] Knig & 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) Questions to answer 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 Outline Original spinless 2 levels x 2 leads Equivalent Anderson model 1 spinful level x 1 ferromagnetic lead Anisotropic Kondo model in a titled magnetic field Use exact solution (Bethe ansatz) Exact mapping Schrieffer-Wolff transformation V = V U >> Observables n 1, n 2, t Isotropic Kondo with a field Inverse mapping, Friedel sum rule The model: notation Two orbital levels Two leads Level spacing h On-site Coulomb U No symmetry imposed on a i ( wide band, D>>U) 0 +h/2 0 h/2 U Singular value decomposition Diagonalize the tunneling matrix: Define new degrees of freedom The pseudo-spin is conserved in tunneling! Singular value decomposition Diagonalize the tunneling matrix: Define new degrees of freedom R d, R l are orthogonal matrices Map onto Anderson scalar spin vector in a tilted magnetic field two preferred directions! Outline Original spinless 2 levels x 2 leads Equivalent Anderson model 1 spinful level x 1 ferromagnetic lead Use exact solution (Bethe ansatz) Exact mapping V = V Observables n 1, n 2, t Inverse mapping, Friedel sum rule Solvable case: isotropic V Standard Anderson: In terms of original couplings: At T=0, an exact solution is possible for n 1, n 2 Numerical solution of Bethe ansatz equations fixed Wiegman (1980); Okiji & Kawakami (1982) one preferred direction Exact results for isotropic AM Friedel-Langreth sum rule: = |V| 2 U n1n2n1n2 n 1 +n 2 1 |t| 2 arg t Glazman & Raikh Local moment single occupancy Polarization charge localization Correlation-driven competition (see later) No phase lapse Outline Original spinless 2 levels x 2 leads Equivalent Anderson model 1 spinful level x 1 ferromagnetic lead Anisotropic Kondo model in a titled magnetic field Exact solution (Bethe ansatz) Exact mapping Schrieffer-Wolff transformation V = V U >> Observables n 1, n 2, t Isotropic Kondo in with a field Inverse mapping, Friedel sum rule 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 Martinek et al., PRL ; (2003) Pasupathy et al., Science 306, 86 (2004) effective Zeeman field Mapping onto a Kondo model Schrieffer-Wolff in CB valley ( U >> , h ) anisotropic exchange effective field Poor mans scaling gives T K Anisotropy is RG irrelevant use results for isotropic Kondo model in Mapping onto a Kondo model Schrieffer-Wolff in CB valley ( U >> , h ) anisotropic exchange effective field Bethe ansatz for isotropic Kondo model by Andrei &Lowenstein (1980) Geometrical interpretation Known function M K Project onto original 1-2 direction Magnetization is determined by the field Transmission L-R: phase shifts via sum rule generalized Glazman-Raikh An example Numbers from Fig.5 of PRL 96, (2006) = 0.97 tot = 0.03 tot d =31 l = 62 Changing gate voltage 0 leads to effective field rotation! SVD angles reflect asymmetry in tunneling U/ tot =3 Small spacing : correlations h=0.01 00 0 = U/2 Small spacing : correlations h tot hh M n 1 -n 2 TKTK |t| 2 Population inversion Silvestrov & Imry (2000) Phase lapse by Silvestrov & Imry (2000) h=0.01 Correlation-induced resonances Meden & Marquardt (2006) h tot hh M n 1 -n 2 00 0 = U/2 |t| 2 h=0.1 Intermediate spacing: rotations ll d +90 Gres et al., PRB 62, 2188(2000) Fano resonances! Occupations numbers and transmission amplitude are always* smooth Generic, sharp -jump of phase for The population inversion and the phase lapse need not to coincide Relevant energy scales Range of 0 -dependent component Transversal projection of level spacing Kondo correlation scale Compare to other methods Both h eff and T K depend on 0 but h = 0 fRG h eff T K => M=1/4 h eff = 0 h eff >T K 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? Kashcheyevs Glazman-Raikh as 2x1 SVD Only one combination couples to the dot Scattering of the coupled mode Langreth (1966) For, unitarity limit VLVL VRVR L R Glazman-Raikh rotation (1988) Example: h=0 (degenerate) h tot hh M n 1 -n 2 00 0 = U/2 TKTK |t| 2 Conductance in isotropic case For h || z, spin is conserved Rotations imply Friedel sum rule 0 /2 - phase shift difference Bethe results An isotropic Kondo model in external field Use exact Bethe ansatz Key quantities Return back Local moment here: