1. Coulomb versus spin-orbit interaction in carbon-nanotube quantum dots Andrea Secchi and Massimo Rontani CNR-INFM Research Center S3 and University of Modena, Modena, Italy exact diagonalization of few-electron Hamiltonian clarification of recent tunneling experiments

2. Carbon-nanotube quantum dots quasi-1D systems double degeneracy F. Kuemmeth et al., Nature 452, 448 (2008)

3. Strong correlation or not in CN QDs?

4. spin-orbit interaction splits 4-fold degenerate spin-orbitals Low temperature SETS experiment spin isospin

5. Strong correlation or not in CN QDs? two-electron ground state: one Slater determinant no correlation chemical potential the simplest interpretation

6. CI model: 1D harmonic potential theory exp configuration-interaction (CI) calculation: two valleys QD: harmonic potential forward & backward Coulomb interactions spin-orbit coupling free parameter:  M. Rontani et al., JCP 124, 124102 (2006)

7. Strongly correlated CI wave functions A & B states: strongly correlated same orbital wave functions differ in isospin only A. Secchi and M. Rontani, arXiv: 0903.5107 isospin = valley population different harmonic oscillator quantum numbers

8. Independent-particle feature explained exp A. Secchi and M. Rontani, arXiv: 0903.5107 N = 2 N = 1 B(T) theo A and B: correlated T 3 = 0, 1 split by spin- orbit int. only T 3 = 0 T 3 = 1 T 3 = 1/2 T 3 = -1/2

9. Non-universal tunneling spectrum exp A. Secchi and M. Rontani, arXiv: 0903.5107 N = 2 N = 1

10. CI two-electron energy spectrum A. Secchi and M. Rontani, arXiv: 0903.5107 gerade ungerade x n(x)n(x)

11. Pair correlation functions g(X) = probability to find a couple of electrons at relative distance X

13. Conclusions spin-orbit and Coulomb interactions coexist non-interacting features of tunneling spectra explained we predict electrons to form a Wigner molecule andrea.secchi@unimore.it massimo.rontani@unimore.it www.s3.infm.it www.nanoscience.unimore.it/max.html

14. Single-particle Hamiltonian Bloch states in K and K’ valleys envelope function spin-orbit interaction and magnetic field

15. Effective 1D Coulomb interaction Ohno potential trace out x and z degrees of freedom forward backward

16. Fully interacting Hamiltonian

17. Spin-orbit coupling for two electrons six-fold degenerate

18. Wigner-Mattis theorem is not appliable in nanotubes nodeless in the ground state S = 0 isospin T = additional degree of freedom either (S = 0, T = 1) or (S = 1, T = 0) T z = -1, 0, +1S z = -1, 0, +1