Phase transition in bilayer quantum Hall system with opposite magnetic field

We construct a mapped bilayer quantum Hall system to realize the proposal that two nearly flatbands have opposite Chern numbers. For the C = ±1 case, the two Landau levels of the bilayer experience opposite magnetic fields. We consider a mapped bilayer quantum Hall system at total filling ${\nu }_{\mathrm{t}}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}$ where the intralayer interaction is repulsive and the interlayer interaction is attractive. We take exact diagonalization (ED) calculations on a torus to study the phase transition when the separation distance d/l_B is driven. The critical point at d_c/l_B = 0.68 is characterized by a collapse of degeneracy and a crossing of energy levels. In the region d/l_B < d_c/l_B, the states of each level are highly degenerate. The pair-correlation function indicates electrons with opposite pseudo-spins are strong correlated at r = 0. We find an exciton stripe phase composed of bound pairs. The ferromagnetic ground state is destroyed by the strong effective attractive potential. An electron composite-Fermion (eCF) and a hole composite Fermion (hCF) are tightly bound. In the region d/l_B > d_c/l_B, a crossover from the d → d_c limit to the large d limit is observed. The electron and hole composite Fermion liquids (CFL) are realized by composite Fermions (CF) which attach opposite fluxes, respectively.