Vortex lattice solutions of the ZHK Chern–Simons equations

We consider the non-relativistic Chern–Simons equations proposed by Zhang, Hansen and Kivelson as the mean field theory of the fractional Hall effect. We prove the existence of the vortex lattice solutions (i.e. solution with lattice symmetry and with topological degree one per lattice cell) similar to the Abrikosov solutions of superconductivity. We derive an asymptotic expression for the energy per unit area and show that it attains minimum at the hexagonal lattice.