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

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

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

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

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

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

Three leads (t2 = t1): t4 t3

1G

3G2G

12

Amplitude ‘leakage’

‘Orbital’ Degeneracy: Orbital Kondo Effect

SU(4) Kondo

SU(4)

Simultaneous screening of charge and spin

Degenerate

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)

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)

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

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

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)

New results using LDECA (comparing with NRG)

1.0

' 1.0

0.125

0.0

U

U

t

t

Density of states

Results with field (12 sites)

How does the Kondo peak behave?

1.0

' 1.0

0.125

0.0

U

U

t

t

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…

No discretization

(ECA)

EXACT

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…)