Phase diagram of $\nu=1/2+1/2$ quantum Hall bilayers

We study $\nu=1/2+1/2$ quantum Hall bilayer systems with different layer distance $d$ by exact diagonalization method. The energy spectrum under periodical and twisted boundary conditions shows the gap opening in spin sector near $d/l_B=1.6$. For the smaller layer distance, we find the the change of low-lying excitations induced by energy level crossing near $d/l_B=1.1$, the goldstone mode or pseudo-spin wave excitation is no longer the lowest excitation. Besides, the drag Hall conductance is finite and non-quantized. For the larger layer distance, we find the intermediate phase with four-fold degeneracy and zero drag Hall conductance, which has distinguished energy spectrum compared with the decoupled CF Fermi liquid states. The transition for different phases has also been discussed.