1. Wigner molecules in carbon-nanotube quantum dots Massimo Rontani and Andrea Secchi S3, Istituto di Nanoscienze – CNR, Modena, Italy

2. ultraclean semiconducting nanotubes Bockrath group, Nature Phys. 2008McEuen group, Nature 2008 gate-defined quantum dots shallow confinement potentials (approx. parabolic)

3. ultraclean semiconducting nanotubes McEuen group, Nature 2008 chemical potential  (N) Bockrath group, Nature Phys. 2008 chemical potential  (N) 0 8 B (T) 1h 3h 5h B (T) 1e 2e 3e

4. ultraclean semiconducting nanotubes Bockrath group, Nature Phys. 2008McEuen group, Nature 2008 chemical potential  (N) 0 8 B (T) 1h 3h 5h B (T) 1e 2e 3e independent from B

5. ultraclean semiconducting nanotubes Bockrath group, Nature Phys. 2008McEuen group, Nature 2008 chemical potential  (N) 0 8 B (T) 1h 3h 5h B (T) 1e 2e 3e spin added electron

6. ultraclean semiconducting nanotubes Bockrath group, Nature Phys. 2008McEuen group, Nature 2008 chemical potential  (N) 0 8 B (T) 1h 3h 5h B (T) 1e 2e 3e isospin added el. (angular momentum)

7. ultraclean semiconducting nanotubes Bockrath group, Nature Phys. 2008McEuen group, Nature 2008 chemical potential  (N) 0 8 B (T) 1h 3h 5h B (T) 1e 2e 3e ground state spin & isospin polarized Wigner molecule? single-particle + spin-orbit

8. motivation Coulomb interaction vs single-particle physics role of interaction? exps at Harvard and Delft on coherent spin manipulation outlook (I) similar issues for graphene quantum dots similar theoretical approach (see next slide)

9. Hamiltonian exact diagonalisation ground & excited states many-body term: Ohno potential, inter- and intra-valley channels (including short range terms) compute the wavefunction as a superposition of Slater determinants Rontani et al., J. Chem. Phys. 124, 124102 (2006) single-particle term: mass + isospin + 1D harmonic confinement + B + spin-orbit coupling compute  (N), n(x), g(x),… envelope function approximation Luttinger and Kohn 1955, Ando 2005

10. experimental evidence split 4-fold degenerate spin-orbitals

11. non-interacting physics? two-electron ground state: one Slater determinant no correlation chemical potential the simplest interpretation

12. theory vs experiment theory PRB 80, 041404(R) (2009)McEuen group 2008 B (T) dielectric constant fitting parameter

13. strongly correlated wave functions A & B states: strongly correlated same orbital wave functions differ in isospin only A. Secchi and M.R., PRB 80, 041404(R) (2009) isospin = valley population

14. spectrum affected by interaction N = 2 N = 1 A. Secchi & M.R., PRB 80, 041404(R) (2009) interaction strength  SO

15. crystallization criterion A. Secchi & M.R., PRB 82, 035417 (2010) Bockrath group, Nature Phys. 2008 chemical potential  (N) 0 8 B (T) 1h 3h 5h

16. crystallization criterion A. Secchi & M.R., PRB 82, 035417 (2010) a = WM b = particle-in-a-box a b

17. conclusions Wigner molecules form in realistic samples outlook (II) quantum devices (localization + spin-orbit coupling + electric control) scanning tunneling spectroscopy www.nanoscience.unimore.it/max.html www.nano.cnr.it nanotube quantum dots strongly correlated graphene quantum dots few-body physics of cold Fermi atoms M. Rontani et al., PRL 102, 060401 (2009)