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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. - PowerPoint PPT Presentation
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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: