Wigner molecules in carbon-nanotube quantum dots

We demonstrate that electrons in quantum dots defined by electrostatic gates in semiconductor nanotubes freeze orderly in space realizing a ``Wigner molecule.'' Our exact diagonalization calculations uncover the features of the electron molecule, which may be accessed by tunneling spectroscopy---indeed some of them have already been observed by Deshpande and Bockrath [Nat. Phys. 4, 314 (2008)]. We show that numerical results are satisfactorily reproduced by a simple ansatz vibrational wave function: electrons have localized wave functions, like nuclei in an ordinary molecule, whereas low-energy excitations are collective vibrations of electrons around their equilibrium positions.