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

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slide1

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)

motivation

Conductance

gate voltage

Phase

Motivation

“Phase lapse”

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

motivation continued
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)

motivation continued1

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
questions to answer
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
outline
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

the model notation

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)

singular value decomposition
Singular value decomposition
  • Diagonalize the tunneling matrix:
  • Define new degrees of freedom
  • The pseudo-spin is conserved in tunneling!
singular value decomposition1
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

Map onto Anderson

two preferred

directions!

outline1
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)

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

Solvable case: isotropic V

one preferred direction

Wiegman (1980); Okiji & Kawakami (1982)

exact results for isotropic am
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

outline2
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

magnetic insights
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)

mapping onto a kondo model
Mapping onto a Kondo model
  • Schrieffer-Wolff in CB valley (U >>Γ, h)

  • anisotropic exchange
  • effective field
mapping onto a kondo model1

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
geometrical interpretation

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:

an example

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!

small spacing correlations

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

intermediate spacing rotations

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

slide23

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
compare to other methods

heff≈TK => M=1/4

fRG

heff >TK

heff >TK

heff= 0

Compare to other methods
  • Both heffand TKdepend on ε0 but h = 0
summary and outlook
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?
slide26

Thanks!

Kashcheyevs

glazman raikh as 2x1 svd
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)

example h 0 degenerate
Example: h=0 (degenerate)

htot

TK

θh

M

n1-n2

|t|2

ε0= – U/2

ε0

conductance in isotropic case

↑-↓ phase shift difference

Conductance in isotropic case
  • For h || z, spin is conserved
  • Rotations imply
  • Friedel sum rule

π/2

0

bethe results

here:

Local moment

Bethe results
  • An isotropic Kondo model in external field
  • Use exact Bethe ansatz
  • Key quantities
  • Return back