We study Skyrmion quantum numbers, charge, and statistics, in $(2+1)$ dimension induced by quadratic band touching (QBT) fermions. It is shown that induced charge of Skyrmions is twice bigger than corresponding Dirac particles’ and their statistics are always bosonic. Applying to the Bernal stacking bilayer graphene, we show that Skyrmions of quantum spin Hall are charge $4e$ bosons, so their condensation realizes charge $4e$ superconductivity. The phase transition could be of second order, and one candidate theory of the transition is an $O\left(5\right)$ nonlinear sigma model with a nonzero Wess-Zumino-Witten term. We calculate the renormalization group beta function of the model perturbatively and propose a possible phase diagram. We also discuss how QBT fermions are different from two copies of Dirac particles.

• Received 10 April 2012

DOI:https://doi.org/10.1103/PhysRevB.85.245123