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 π 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) ε1ε1 ε2ε2

Motivation continued U Non-monotonic level filling and 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) ε1ε1 ε2ε2 Two orbital levels Two leads On-site repulsion U Spinless electrons

Non-monotonic level filling and 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) 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 >> Γ 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 αi ( wide band, D>>U) ε 0 +h/2 ε 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: Γ Γ=πρ |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 >> Γ 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 >> Γ, h ) … –anisotropic exchange –effective field

Poor man’s 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 >> Γ, 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) Γ ↑ = 0.97 Γ tot Γ ↓ = 0.03 Γ tot θ d =31º θ l = 62º Changing gate voltage ε 0 leads to effective field rotation! SVD angles reflect asymmetry in tunneling U/Γ tot =3

Small spacing : correlations h=0.01

ε0ε0 ε 0 = – U/2 Small spacing : correlations h tot θhθh M n 1 -n 2 TKTK |t| 2 Population inversion Silvestrov & Imry (2000)  Phase lapse by π Silvestrov & Imry (2000) h=0.01 “Correlation-induced resonances” Meden & Marquardt (2006)

h tot θhθh M n 1 -n 2 ε0ε0 ε 0 = – U/2 |t| 2 h=0.1 Intermediate spacing: rotations θlθl θ d +90º Göres et al., PRB 62, 2188(2000) Fano resonances!

Occupations numbers and transmission amplitude are always* smooth Generic, sharp π -jump of phase for The population inversion and the phase lapse need not to coincide Relevant energy scales Range of ε 0 -dependent component Transversal projection of level spacing Kondo correlation scale

Compare to other methods Both h eff and T K depend on ε 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 θhθh M n 1 -n 2 ε0ε0 ε 0 = – U/2 TKTK |t| 2

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

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