Wigner crystal phases in confined carbon nanotubes

Abstract

We present a detailed theoretical analysis of the Wigner crystal states in confined semiconducting carbon nanotubes. We show by detailed semimicroscopic calculations that the effective exchange interaction has an SU(4) symmetry, and can reach values as large as $J\ensuremath{\sim}100\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ in weakly screened, small diameter nanotubes, close to the Wigner crystal---electron liquid crossover. This large value of the exchange coupling in the crossover region also follows from robust scaling arguments. Modeling the nanotube carefully and analyzing the magnetic structure of the inhomogeneous electron crystal, we recover the experimentally observed ``phase boundaries'' of Deshpande and Bockrath [V. V. Deshpande and M. Bockrath, Nat. Phys. 4, 314 (2008)]. Spin-orbit coupling only slightly modifies these boundaries, but breaks the spin symmetry down to $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$, and in Wigner molecules it gives rise to interesting excitation spectra, reflecting the underlying SU(4) as well as the residual $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetries.