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![Page 1: Application of the Cluster Embedding Method to Transport Through Anderson Impurities George Martins Carlos Busser Physics Department Oakland University.](https://reader035.fdocuments.net/reader035/viewer/2022062517/56649f115503460f94c23c43/html5/thumbnails/1.jpg)
Application of the Cluster Embedding Method to Transport Through Anderson Impurities
George MartinsCarlos Busser
Physics DepartmentOakland University
Enrique Anda and Maria Davidovich (Puc – Rio)Guillermo Chiappe (Alicante)Elbio Dagotto (Oak Ridge)Adrian Feiguin (Project Q – Microsoft)Fabian Heidrich-Meisner (Aachen)
Materials World
Network
ColaboracionInteramericana
de Materiais
A method to study highly correlated nanostructures
Workshop on Decoherence, Correlations and Spin Effects on Nanostructured Materials – Vina del Mar – Chile 2009
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Triangular geometry: interference and amplitude leakage
Enrique Anda (PUC – Rio)Carlos Busser (Oakland)Nancy Sandler and Sergio Ulloa (Ohio)Edson Vernek (Uberlandia)
3 4 1 2, ,t t U t t
Treat the 3 dots as a molecule
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Bonding, non-bonding and anti-bonding orbitals
1 2 2A B C
2 2A C
3 2 2A B C
1 2 6A B C
3 3A B C
2 2A C
4 0t 3 QDs in series
4 3t tequilateral
A B C 1 2 3
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gV
Just two leads (t2 = 0): t4 t3
2e4e
6e
gV
1
2
3
1
3
1.0
0.45
0.5
U
t
t
Conductance: LDECA (blue) and Finite U Slave bosons (red)
interference
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The ‘partial’ conductances
1
2
3
3Lt
2
223 3 3L L R F
eG t G
h
3RGL
int 1 2 1 2 122 cosG G G GG
2112
2 1
ln rr
r r
GGi
G G
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Three leads (finite t and new parameter values)
A
BC
2A B C 2A B C
A C
1
3
1.0
0.45
0.5
U
t
t
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Three leads (t2 = t1): t4 t3
1G
3G2G
12
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Amplitude ‘leakage’
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‘Orbital’ Degeneracy: Orbital Kondo Effect
SU(4) Kondo
SU(4)
Simultaneous screening of charge and spin
Degenerate
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Model and Hamiltonian
, ; 2d g
UH n n V n U n n
int 0, ; ;
h. c.ll L R
H t d c
t t t
t t t
0t
0t0t
0t
U
U
SU(4)
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Spin-charge ‘entanglement’
Schematics of a co-tunneling process for the usual spin SU(2) Kondo screening.
Same as above, but now for an orbital degree of freedom(orbital SU(2) Kondo).
Simultaneous screening of orbital and spin degrees of freedom, leading to SU(4) Kondo.
P. J. – Herrero et al., Nature 434, 484 (2005)
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SU(4) at Half-filling and NFL Behavior ECA Results
U U SU(4)
Galpin, Logan, and KrishnamurthyPRL 94, 186406 (2005)
-0.04 -0.02 0.00 0.02 0.04
LD
OS
U'=0.0 U'=0.2 U'=0.3 U'=0.4 U'=0.5
-0.04 -0.02 0.00 0.02 0.04
LD
OS
()
U'=0.5 U'=0.6 U'=0.7 U'=0.8
0.0 0.2 0.4 0.6 0.8
2.05
2.10
2.15
2.20
2.25
2.30
2.35
2.40
2.45
(10
-2)
U'
U U SU(4)
-1.6 -1.2 -0.8 -0.4 0.00.00.20.40.60.81.01.21.41.61.82.0
U'=0.0U=0.5t'=0.2t"=0.0B=0.0
G(2
e2 /h) a
nd
<n
>
Vg
U'=0.5 U U
SU(2)
CO
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Conductance Results
-2.0 -1.5 -1.0 -0.5 0.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
-1.5 -1.0 -0.5 0.0 0.5
G(2
e2 /h
)
Vg
U=U'=0.5E=0.035t'=0.2
t'=0.1
Vg
E
2n3n
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Magnetic Field Dependence
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.00.0 0.5 1.0 1.5 2.0 2.5 3.0
U=U'=0.5t'=0.2t"=0.0
Esp
Esp
B
Vg E
orb
orb
=0.2
sp
=0.04
a
-2.5
-2.0
-1.5
-1.0
-0.5 0.0
0.5
1.0 0.
00.
51.
01.
52.
02.
53.
0
U=U
'=0.
5t'=
0.2
t"=0
.0
E
sp
Esp
B
Vg
E
orb
orb=0
.2
sp=0
.04
a
-1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0 t"=0.0 t"=0.05 t"=0.1 t"=0.15 t"=0.175 t"=0.2
G(2
e2 /h)
Vg
U=U'=0.5t'=0.2B=0.0
SU(4) to 2LSU(2)
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New results using LDECA (comparing with NRG)
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1.0
' 1.0
0.125
0.0
U
U
t
t
Density of states
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Results with field (12 sites)
How does the Kondo peak behave?
1.0
' 1.0
0.125
0.0
U
U
t
t
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LDOS with field (half-filling)
The peak seems to split at any finite field.
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Closer view
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Conclusions
New numerical results for conductance in Carbon Nanotubes were presented
ECA method seems capable of capturing glimpses of NFL behavior suggested by previous NRG results
SU(4) regime at half-filling (HF) is confirmed: conductance results for third shell may then be reinterpreted as signature of SU(4) at HF
Calculations at finite magnetic field agree quite well with experimental results
Results indicating how conductance changes from SU(4) to 2LSU(2) regime were presented
More detailed results with field seem to indicate that Kondo peak splits for any finite field.
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DMRG: the future of LDECA?
Currently, the method is based on using Lanczos to solve for the Green’s functions of the cluster.• Advantage: Lanczos is fast and easy to program• Disadvantage: Maximum cluster size is still
small. Finite size effects may occur.
Solution? Use DMRG instead of Lanczos• Advantage: REALLY Larger clusters• Disadvantage: CPU time.
Accuracy of Green’s functions?
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Size (only) doesn’t matter…
No discretization
(ECA)
EXACT
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The importance of being discrete…
LDECA
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Conclusions
An improvement of embedding method was presented Results for single quantum dot agree perfectly with
Bethe ansatz Results for density of states agree with NRG Two stage Kondo system (two hanging quantum dots)
was discussed and compared with NRG Triangular configuration analyzed (interference) SU(4) in carbon nanotubes was analyzed Preliminary results using DMRG instead of Exact
Diagonalization (very encouraging!) For the future:
• Use two-particle Green’s function to calculate embedded spin correlations
• Add temperature and bias (ambitious…)