Coulomb interactions and delocalization in quantum Hall constrictions

Abstract

We study a geometry-dependent effect of long-range Coulomb interactions on quantum Hall (QH) tunneling junctions. In an X-shaped geometry, duality relates junctions with opening angles $\ensuremath{\alpha}$ and $(\ensuremath{\pi}\ensuremath{-}\ensuremath{\alpha}).$ We prove that duality between weak tunneling and weak backscattering survives in the presence of long-range interactions, and that their effects are precisely cancelled in the self-dual geometry $\ensuremath{\alpha}=\ensuremath{\pi}/2.$ Tunneling exponents as a function of $\ensuremath{\alpha},$ the interaction strength $\ensuremath{\chi},$ and the filling fraction $\ensuremath{\nu}$ are calculated. We find that Coulomb interaction induces localization in narrow channels (large $\ensuremath{\alpha}),$ and delocalization for sharply pinched constrictions (small $\ensuremath{\alpha}).$ Consequently, an insulator-to-metal transition happens at an angle ${\ensuremath{\alpha}}_{c}(\ensuremath{\chi},\ensuremath{\nu})<~\ensuremath{\pi}/2.$ We discuss the implications of our results for tunneling experiments in QH-constriction and cleaved-edge geometries.