Capacitance of a resonant level coupled to Luttinger liquids

In this Rapid Communication, we study the differential capacitance of a single-level quantum dot attached to quantum wires in the Luttinger-liquid phase or to fractional quantum-Hall edges, by both tunneling and interactions. We show that logarithmic or even power-law divergence of the capacitance at low temperatures may result in a substantial region of the parameter space (including the v=1/3 filling). This behavior is a manifestation of generalized two-channel Kondo physics and can be used to extract Luttinger-liquid parameters from thermodynamic measurements.