Application of the Cluster Embedding Method to Transport Through Anderson Impurities

1. Application of the Cluster Embedding Method to Transport Through Anderson Impurities A method to study highly correlated nanostructures George Martins Carlos Busser Physics Department Oakland University Colaboracion Interamericana de Materiais Materials World Network 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

2. Triangular geometry: interference and amplitude leakage Treat the 3 dots as a molecule Enrique Anda (PUC – Rio) Carlos Busser (Oakland) Nancy Sandler and Sergio Ulloa (Ohio) Edson Vernek (Uberlandia)

3. Bonding, non-bonding and anti-bonding orbitals 3 QDs in series equilateral

4. Just two leads (t2 = 0): t4 t3 Conductance: LDECA (blue) and Finite U Slave bosons (red) interference

5. The ‘partial’ conductances

6. Three leads (finite t and new parameter values)

7. Three leads (t2 = t1): t4 t3

8. Amplitude ‘leakage’

9. SU(4) ‘Orbital’ Degeneracy: Orbital Kondo Effect SU(4) Kondo Simultaneous screening of charge and spin Degenerate

10. Model and Hamiltonian SU(4)

11. 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.

12. CO SU(4) at Half-filling and NFL Behavior ECA Results Galpin, Logan, and Krishnamurthy PRL 94, 186406 (2005) SU(4) SU(4) SU(2)

13. Conductance Results

14. Magnetic Field Dependence SU(4) to 2LSU(2)

15. New results using LDECA (comparing with NRG)

16. Density of states

17. Results with field (12 sites) How does the Kondo peak behave?

18. LDOS with field (half-filling) The peak seems to split at any finite field.

19. Closer view

20. 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.

21. 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?

22. Size (only) doesn’t matter… EXACT No discretization (ECA)

23. The importance of being discrete… LDECA

24. 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…)