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

Abstract

Chiral spin liquids realize a topological state whose universal response is a fractional spin Hall conductance $ν_s$. 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 $1/L$ 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 $J$--$J_χ$ Hamiltonian ($ν_s=-0.500\pm0.011$). All three agree with the analytic prediction without adjustable parameters, providing a fully quantitative bridge between microscopic topology and observable fractional response.