  The experimental discovery of the fractional Hall conductivity in
two-dimensional electron gases revealed new types of quantum particles, called
anyons, which are beyond bosons and fermions as they possess fractionalized
exchange statistics. These anyons are usually studied deep inside an insulating
topological phase. It is natural to ask whether such fractionalization can be
detected more broadly, say near a phase transition from a conventional to a
topological phase. To answer this question, we study a strongly correlated
quantum phase transition between a topological state, called a $\mathbb{Z}_2$
quantum spin liquid, and a conventional superfluid using large-scale quantum
Monte Carlo simulations. Our results show that the universal conductivity at
the quantum critical point becomes a simple fraction of its value at the
conventional insulator-to-superfluid transition. Moreover, a dynamically
self-dual optical conductivity emerges at low temperatures above the transition
point, indicating the presence of the elusive vison particles. Our study opens
the door for the experimental detection of anyons in a broader regime, and has
ramifications in the study of quantum materials, programmable quantum
simulators, and ultra-cold atomic gases. In the latter case, we discuss the
feasibility of measurements in optical lattices using current techniques.
