$\nu=1$ bilayer in a periodic potential

The clean $\nu = 1$ quantum Hall bilayer is an excitonic superfluid. Experimentally, due to disorder, the counterflow conductivity $\sigma_{CF}$ remains finite even at the lowest $T$ and the zero-bias peak has finite width. We mimic the nonperturbative effects of disorder [1] by a periodic potential[2] which couples, in a spin-only model, to the topological density ${\mathbf n} \cdot{\partial_x{\mathbf n}}\times{\partial_y{\mathbf n}}$(=charge density). We find a set of ground state phase transitions as the potential strength increases, with increasing local charge density. The transitions are weakly first order, with a new, quadratically dispersing, charge-carrying mode which represent incipient meron-antimeron pairs forming in regions of large potential gradient. These modes can become nearly gapless at and near the transition, which we argue leads to a strong suppression of the interlayer tunneling h. We demonstrate that near the transitions vortex-antivortex pairs become easy to create, leading to a strong suppression of $T_{KT}$ . We discuss an effective theory that incorporates both the Goldstone mode and the new, quadratically dispersion mode. \\[4pt] [1] H. A. Fertig and G. Murthy, prl {\bf 95}, 156802 (2005).\\[0pt] [2] G. Murthy and S. Sachdev, prl {\bf 101}, 226801 (2008).