From Dirac Cones to Semions: An Exact Finite-Size Theory of Parity-Anomaly Transport in Chiral Spin Liquids

Chiral spin liquids realize a topological state whose universal response is a fractional spin Hall conductance ${\nu }_s$. The three quantities that determine this response, the integer Chern number of the fractionalized spinons, the level of the emergent Chern–Simons gauge field, and the physically measured spin pump, are related but distinct, and their relation is often stated only schematically. Here we derive it from a single object: the parity-odd determinant of a gapped Dirac cone on a spatial cylinder, resummed exactly to all orders in the compact holonomy. This determinant fixes the map from spinon topology to measurable response, and proves that finite-size corrections to the topological pump are strictly exponential, with no universal $1\mathrm{/}L$ term. We test the resulting predictions on the kagome chiral spin liquid at three independent levels: the exact one-loop field theory, a parton band-structure calculation ($C=-1$, converging exponentially over cylinders four to twelve sites wide), and an interacting density-matrix renormalization group flux pump on the explicitly chiral $J$–$J_{\chi }$ Hamiltonian (${\nu }_s=-0.500\pm 0.011$). All three agree with the analytic prediction without adjustable parameters, providing a fully quantitative bridge between microscopic topology and observable fractional response.