Slava KashcheyevsAvraham SchillerAmnon AharonyOra Entin-Wohlman

Interference and correlations in two-level dots

Phys. Rev. B 75, 115313 (2007)

Also: Silvestrov & Imry, PRB 75, 115335 (2007)

Lee & Kim, PRL 98, 186805 (2007)

gate voltage

Con

duct

ance

Motivation

Avinun-Kalish et al.,Nature 436 (2005)Schuster et al., Nature 385 (1997)

Pha

se“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)

ε1

ε2

Motivation continued

U

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)

ε1

ε2

• Two orbital levels• Two leads

• On-site repulsion U• Spinless electrons

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)

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

OutlineOriginal 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

Observablesn1, n2, 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 aαi

(wide band, D>>U)

ε0+h/2

ε0–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

• Rd, Rl are orthogonal matrices

Map onto Anderson

scalar

spin vector in a tilted magnetic field

two preferreddirections!

OutlineOriginal 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↓

Observablesn1, n2, 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 n1, n2

• Numerical solution of Bethe ansatz equations

fixed

Wiegman (1980); Okiji & Kawakami (1982)

one preferred direction

Exact results for isotropic AM

Friedel-Langrethsum rule:

Γ Γ=πρ|V|2

Un1

n2

n1+n2 ≈ 1

|t|2

arg t

Glazman & Raikh

• Local moment single occupancy

• Polarization charge localization

• Correlation-driven competition (see later)

• No phase lapse

OutlineOriginal 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

Observablesn1, n2, 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 91 127203; 247202 (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 TK

• 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 modelby Andrei &Lowenstein (1980)

Geometrical interpretation

• Known function MK

• Project onto original1-2 direction

• Magnetization is determined by the field

Transmission L-R:

phase shifts via sum rulegeneralized

Glazman-Raikh

An exampleNumbers from Fig.5 of PRL 96, 146801 (2006)

Γ↑ = 0.97 Γtot

Γ↓ = 0.03 Γtot

θd=31º

θl = 62º

Changing gate voltage ε0

leads to effective field rotation!

SVD angles reflect asymmetry in tunneling

0.47 0.25

0.08 0.16

U/Γtot =3

Small spacing : correlations

h=0.01

h=0.01

ε0

ε0= – U/2

Small spacing : correlations

htot

θh

M

n1-n2

TK

|t|2

Population inversionSilvestrov & Imry (2000)

Phase lapse by πSilvestrov & Imry (2000)

h=0.01“Correlation-induced

resonances”Meden & Marquardt (2006)

htot

θh

M

n1-n2

ε0

ε0= – U/2

|t|2h=0.1

Intermediate spacing: rotations

θl θd+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 ε0-dependent component

• Transversal projection of level spacing

• Kondo correlation scale

Compare to other methods• Both heff and TK depend on ε0 but h = 0

fRG

heff ≈ TK => M=1/4

heff = 0

heff >TK heff >TK

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”

VL VRL R

Glazman-Raikh rotation (1988)

Example: h=0 (degenerate)htot

θh

M

n1-n2

ε0

ε0= – U/2

TK

|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: