Numerical Study of Quantum Hall Bilayers at Total Filling $\nu_T=1$: A New Phase at Intermediate Layer Distances

We study the phase diagram of quantum Hall bilayer systems with total filing $\nu_T=1/2+1/2$ of the lowest Landau level as a function of layer distances $d$. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phase with spontaneous interlayer coherence at small $d$, a composite Fermi liquid at large $d$, and an intermediate phase for $1.1<d/l_B<1.8$ ($l_B$ is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i) a dramatic change in the Berry curvature of the ground state under twisted boundary conditions on the two layers; (ii) an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by the vanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling $\nu_T=1$.