Bound states of anyons: a geometric quantization approach

Abstract

The question of anyon interactions and their possible binding plays a key role in the physics of fractional quantum Hall states. Here, we introduce a controlled and scalable approach to study anyon binding by working entirely within the Hilbert space of anyons. The resulting theory is characterized by an effective potential, which captures the electrostatic energy of classical anyon configurations, and a K\"ahler potential, which simultaneously encodes the anyon Berry phase and the structure of their Hilbert space; both quantities are readily computed using Monte Carlo methods for large systems, enabling reliable extrapolation to the thermodynamic limit. By applying the formalism of geometric quantization on K\"ahler manifolds, we construct the anyon Hamiltonian, which can be exactly diagonalized in the few-anyon Hilbert space. Applying our approach to the quasiholes of the $\nu=1/3$ Laughlin state with screened Coulomb interaction, we find that Laughlin quasiholes form bound states for screening lengths comparable or smaller than the magnetic length. Remarkably, binding occurs despite both the bare electron-electron interaction and the quasihole electrostatic potential being purely repulsive. The bound-state formation is a Berry phase effect, driven by oscillations in the quasihole density profile on the $\ell_B$ scale that are invisible in the quasihole electrostatic potential alone. For multiple quasiholes, we identify a sequence of phases as the screening length is reduced: free $e/3$ anyons, paired $2e/3$ bound states, three-anyon charge-$e$ clusters, and larger composite objects. Finally, we discuss possible signatures in charge imaging experiments on quantum Hall systems and the relevance to the phase diagram of itinerant anyon phases in fractional quantum anomalous Hall materials.