Application of the Cluster Embedding Method to
Transport Through Anderson Impurities
A method to study highly correlated nanostructures
Materials
World
Network
George Martins
Carlos Busser
Physics Department
Oakland University
Colaboracion
Interamericana
de Materiais
Enrique Anda and Maria Davidovich (Puc – Rio)
Guillermo Chiappe (Alicante)
Elbio Dagotto (Oak Ridge)
Adrian Feiguin (Project Q – Microsoft)
Fabian Heidrich-Meisner (Aachen)
Workshop on Decoherence, Correlations and Spin Effects
on Nanostructured Materials – Vina del Mar – Chile 2009
Triangular geometry: interference and
amplitude leakage
t3 , t4  U  t1 , t2
Treat the 3 dots
as a molecule
Enrique Anda (PUC – Rio)
Carlos Busser (Oakland)
Nancy Sandler and Sergio Ulloa (Ohio)
Edson Vernek (Uberlandia)
Bonding, non-bonding and antibonding orbitals
t4  0
3 QDs in
series
 1   A  2 B  C  2
 2   A  C  2
 3   A  2 B  C  2
A B C 12 3
t4  t3
equilateral
 1   A  2 B  C 
 2   A  C 
6
2
 3   A  B  C 
3
Just two leads (t2 = 0): t4  t3
Conductance: LDECA (blue) and Finite U Slave bosons (red)

4e  2e
3
6e
U  1.0
t1  0.45

1
2
Vg
Vg
t3  0.5
interference
The ‘partial’ conductances
3
2
L
2
e2
2
G3   Lt L 3 G3 R   F 
h
Gint  G1  G2  2 G1G2 cos 12
 G1r G2 r 
i12  ln 

G
G
 2 r 1r 
tL3
1
G3R
Three leads (finite t and new parameter
values)
A
C
B
A C
U  1.0
t1  0.45
t3  0.5
A  2 B C
A  2 B C
Three leads (t2 = t1): t4  t3
G2
G3
G1
12
Amplitude ‘leakage’
‘Orbital’ Degeneracy: Orbital Kondo
Effect
SU(4) Kondo
Simultaneous
screening of
charge and spin
Degenerate
SU(4)
Model and Hamiltonian
U
SU(4)
U
U

H d    n n  Vg n   U  n n 

  ,  ;  2
 

H int    t  d
cl 0  h. c.
l  L , R  ; ;
t  0
t   0
t  t  t 
t  t  t 
t  0
t   0
Spin-charge ‘entanglement’
P. J. – Herrero et al., Nature 434, 484 (2005)
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.
U=0.5
t'=0.2
t"=0.0
B=0.0
-1.6
U'=0.0
U'=0.2
U'=0.3
U'=0.4
U'=0.5
U'=0.0
U'=0.5
-1.2
-0.8
Vg
-0.4
-0.04
0.0
-0.02
U
0.00

0.02
Galpin, Logan, and Krishnamurthy
PRL 94, 186406 (2005)
-2
SU(4)
-0.04
0.04
2.25
2.20
2.15
0.0
CO
0.00

0.02
2.35
2.30
U  U
SU(4)
2.10
2.05
SU(2)
-0.02
U
2.45
2.40
(10 )
U  U
U'=0.5
U'=0.6
U'=0.7
U'=0.8
LDOS()
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
LDOS 
G(2e2/h) and <n>
SU(4) at Half-filling and NFL Behavior ECA
Results
0.2
0.4
U'
0.6
0.8
0.04
Conductance Results
2.0
1.8
t'=0.2
1.6
G(2e2/h)
1.4
1.2
E
U=U'=0.5
E=0.035
t'=0.1
1.0
0.8
0.6
0.4
0.2
n3
0.0
-2.0 -1.5 -1.0 -0.5
Vg
n2
0.0
-1.5 -1.0 -0.5
Vg
0.0
0.5
U=U'=0.5
-2.0 t'=0.2
-1.5 t"=0.0
-0.5
Esp
G(2e2/h)
B
1.5
1.0
B
Esp
1.6
2.0
1.2
1.0
0.8
0.5
0.6
2.5
1.0
0.0
1.5
0.0
1.0
-1.0
-2.5
0.5
Vg
1.0
0.0
Esp
0.5  =0.04
sp
0.5  =0.04
sp
orb=0.2
orb=0.2
U=U'=0.5
-2.0 t'=0.2
-1.5 t"=0.0
0.0
t"=0.0
t"=0.05
t"=0.1
t"=0.15
t"=0.175
t"=0.2
1.4
Eorb
-0.5
2.0
1.8
Eorb
Vg
-1.0
a
2.0
-2.5
SU(4) to 2LSU(2)
2.5
Esp
a
3.0
Magnetic Field Dependence
3.0
0.4
0.2
U=U'=0.5
t'=0.2
B=0.0
0.0
-1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25
Vg
0.00
0.25
New results using LDECA (comparing
with NRG)
Density of states
U  1.0
U '  1.0
t   0.125
t   0.0
Results with field (12 sites)
U  1.0
U '  1.0
t   0.125
t   0.0
How does the Kondo peak behave?
LDOS with field (half-filling)
The peak seems
to split at any
finite field.
Closer view
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.
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?
Size (only) doesn’t matter…
EXACT
No
discretization
(ECA)
The importance of being discrete…
LDECA
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…)