Resonantly interacting $2e/3$ anyons: a controlled microscopic theory of anyon superconductivity

Abstract

We develop a microscopic theory of anyon superconductivity that connects electronic interactions in a lightly doped Laughlin state to the universal physics of a resonantly interacting anyon gas. Using geometric quantization of Laughlin quasiholes, we derive the effective many-body Hamiltonian of dilute charge-$2e/3$ anyons, including their valley structure, dispersion, and statistical and short-range interactions, together with an explicit map between anyon and electron observables. This reduces the dilute many-body problem to a few parameters fixed by data computable from few anyon numerics. In analogy with resonant Fermi and Bose gases, we organize the dilute anyon problem around a two-body binding resonance and its relevant and marginal perturbations. For a three-valley gas of charge-$2e/3$ anyons, we identify a resonance whose long-distance theory is governed by an Aharonov-Casher anyon Hamiltonian whose magnetic field is the statistical Chern-Simons field and whose ground state is exactly solvable. Using the ground state wavefunction, we demonstrate Cooper-pair off-diagonal long-range order directly in electron coordinates and determine the pairing symmetry and vortex-lattice structure. The leading perturbations at the fixed point are a relevant pair detuning and a marginal three-body contact in the valley-singlet channel, which determine the compressibility and sound velocity. A repulsive three-body coupling is marginally irrelevant, allowing the fixed point to control a parametrically broad dilute regime near pair resonance, while attractive three-body interactions instead generate charge-$2e$ trimers. More broadly, our results show that binding thresholds between competing anyon composites host universal resonant regimes that are not captured by the lowest-energy anyon species alone, providing a new setting for controlled theories of anyon matter.