Unveiling the Interplay between Chiral Quantum Phases and Geometry Fluctuations: the Superfluid Transition on a Membrane

Chiral quantum phases couple to the background geometry, which leads to intriguing effects such as the gravitational anomaly in fractional quantum Hall effect and geometric induction in chiral condensates. However, the impact of geometry fluctuations at finite temperatures remains relatively uncharted. In this study, we investigate the behavior of a chiral superfluid situated on a flexible two-dimensional membrane. We show that the Berezinskii-Kosterlitz-Thouless (BKT) transition is renormalized by geometry fluctuations, resulting in a critical temperature depending on the bending rigidity of the membrane. Conversely, the presence of a superfluid significantly alters the long-distance behavior of the bending rigidity and the associated crumpling transition. Using a renormalization group treatment of the continuum field theory describing the coupling of superfluid phase to the flexural and in-plane phonons, we derive the joint phase diagrams for the cases of a fluid, hexatic, and crystalline membrane, with and without external tension.