Transcript Interference and correlations in two

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
Motivation
“Phase lapse”
Phase
gate voltage
Avinun-Kalish et al.,Nature 436 (2005)
Schuster et al., Nature 385 (1997)
Motivation continued
ε1
U
ε2
Explicit on-site
Coulomb interaction
Interaction-based qualitative explanation
of the phase lapse universality:
Silvestrov & Imry PRL 85 (2000)
• Destructive
interference – several
paths through the dot
• Non-interacting model
gives either 0 or π
phase change
between the
resonances
Entin-Wohlman, Hartzstein & Imry (1986)
Silva, Oreg & Gefen (2002)
Entin-Wohlman,Aharony,
Levinson&Imry (2002)
Motivation continued
ε1
 Non-monotonic level filling
and population inversion
U
– Silvestrov & Imry (2000) [mechanism & PT]
– König & Gefen PRB 71 (2005)
[perturbation in tunneling]
– Sindel, Silva, Oreg & von Delft
PRB 72 (2005) [NRG & Hartree-Fock]
ε2
• Two orbital levels
• Two leads
 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)]
• On-site repulsion U
• Spinless electrons
– Karrasch,Hecht,Weichselbaum,Oreg,
vonDelft & Meden PRL(2007)
[NRG & fRG]
 Orbital Kondo physics
(“Correlation-induced” resonances)
Questions to answer
 Accurate methods…
 Non-monotonic level filling
and population inversion
– either numrical only
– or too narrow validity range
– Silvestrov & Imry (2000) [mechanism & PT]
– König & Gefen PRB 71 (2005)
[perturbation in tunneling]
 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
– 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)
Outline
Original spinless
2 levels x 2 leads
Exact mapping
Inverse mapping,
Friedel sum rule
Equivalent Anderson model
1 spinful level x
1 ferromagnetic lead
Schrieffer-Wolff
transformation
Observables
n1, n2, t
V↑ = V↓
Use exact solution
(Bethe ansatz)
U >> Γ
Anisotropic Kondo model
in a titled magnetic field
Isotropic Kondo
with a field
The model: notation
ε0+h/2
U
ε0–h/2
• Two orbital levels
• Two leads
• Level spacing h
• On-site Coulomb U
• No symmetry
imposed on aαi
(wide band, D>>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
two preferred
directions!
scalar
spin vector in a tilted magnetic field
Outline
Original spinless
2 levels x 2 leads
Exact mapping
Equivalent Anderson model
1 spinful level x
1 ferromagnetic lead
Observables
n1, n2, t
Inverse mapping,
Friedel sum rule
V↑ = V↓
Use exact solution
(Bethe ansatz)
Solvable case: isotropic V
• “Standard” Anderson:
• In terms of original couplings:
fixed
one preferred direction
• At T=0, an exact solution is possible for n1, n2
• Numerical solution of Bethe ansatz equations
Wiegman (1980); Okiji & Kawakami (1982)
Exact results for isotropic AM
• Local moment 
single occupancy
• Polarization
n1+n2 ≈ 1
charge localization
• Correlation-driven
competition (see later)
Γ=πρ|V|2
Γ
U
n1
n2
|t|2
arg t
• No phase lapse
Friedel-Langreth
sum rule:
Glazman
& Raikh
Outline
Original spinless
2 levels x 2 leads
Exact mapping
Inverse mapping,
Friedel sum rule
Equivalent Anderson model
1 spinful level x
1 ferromagnetic lead
Schrieffer-Wolff
transformation
Observables
n1, n2, t
V↑ = V↓
Exact solution
(Bethe ansatz)
U >> Γ
Anisotropic Kondo model
in a titled magnetic field
Isotropic Kondo in
with a field
Magnetic insights…
• A quantum dot with ferromagnetic leads
– V↑ ≠ V↓ generates additional local field
– the physics: renormalization of level positions
Martinek et al., PRL 91
127203; 247202 (2003)
effective
Zeeman field
Pasupathy et al.,
Science 306, 86 (2004)
• 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
Mapping onto a Kondo model
• Schrieffer-Wolff in CB valley (U >> Γ, h)
…
– anisotropic exchange
– effective field
Mapping onto a Kondo model
• Schrieffer-Wolff in CB valley (U >> Γ, h)
…
• Poor man’s scaling gives TK
– anisotropicisexchange
• Anisotropy
RG irrelevant
–
field
– effective
use results
for isotropic Kondo model in
Geometrical interpretation
• Magnetization is
determined by the field
• Known function MK
• Project onto original
1-2 direction
generalized
phase shifts via sum rule
Transmission L-R:
Glazman-Raikh
Bethe ansatz for isotropic Kondo model
by Andrei &Lowenstein (1980)
An example
Numbers from Fig.5 of
PRL 96, 146801 (2006)
θd=31º
0.47
0.25
U/Γtot =3
θl = 62º
0.08
SVD angles reflect
asymmetry in tunneling
Γ↑ = 0.97 Γtot
Γ↓ = 0.03 Γtot
Changing gate voltage ε0
leads to effective field rotation!
0.16
Small spacing : correlations
h=0.01
Small spacing : correlations
TK
htot
θh
M
Population inversion
Silvestrov & Imry (2000)
n1-n2
“Correlation-induced
h=0.01
h=0.01 resonances”
Meden & Marquardt (2006)
2
|t|

Phase lapse by π
Silvestrov & Imry (2000)
ε0= – U/2
ε0
Intermediate spacing: rotations
h
Görestot
et al., PRB
62, 2188(2000)
θl
θd+90º θh
M
Fano resonances!
n1-n2
|t|2
h=0.1
ε0= – U/2
ε0
Relevant energy scales
• Range of ε0-dependent component
• Transversal projection of level spacing
• Kondo correlation scale
 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
Compare to other methods
• Both heff and TK depend on ε0 but h = 0
heff ≈ TK => M=1/4
fRG
heff >TK
heff = 0
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
L
VL
VR
R
• Only one combination
couples to the dot
• Scattering of the
coupled mode
• Langreth (1966)
• For
Glazman-Raikh rotation (1988)
,
“unitarity limit”
Example: h=0 (degenerate)
TK
htot
θh
M
n1-n2
|t|2
ε0= – U/2
ε0
Conductance in isotropic case
• For h || z,
spin is conserved
• Rotations imply
↑-↓ phase shift difference
• Friedel sum rule
0
π/2
Bethe results
• An isotropic
Kondo model in
external field
• Use exact
Bethe ansatz
• Key quantities
• Return back
Local moment
here: