Vortex Solutions of Bilayer Quantum Hall Systems at υ=1/2

We investigate the static vortex solutions of a bilayer quantum Hall state at the Landau-level filling factor υ = 1/2. This work is based on the ZHK model, which is an effective field theory including ChernSimons gauge interactions. We deduce the dimensionless nonlinear equations of motion for vortices possessing cylindrically symmetry, and analyze the asymptotical behaviors of solutions. Additionally, we analyze the values of critical coupling constants under the self-dual condition, and obtain the self-dual equations. Finally, vortices of type (0, 1), (0, −1), (1, −1) and (−1, −1) are solved with numerical methods. We reach the conclusion that vortex of type (1, −1) is unstable, which will decay to (1, 0) and (0, −1). The vortices of type (0, −1) and (−1, −1) are self-dual solutions from numerical results.