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

Abstract

Chiral spin liquids carry a hidden bookkeeping problem: the integer Chern number of their fractionalized spinons, the level of the emergent Chern--Simons gauge field, and the fractional spin response actually measured in experiment or simulation are related but distinct quantities, and the literature routinely conflates them. Here we resolve this by deriving the exact parity-odd determinant of a gapped Dirac cone on a spatial cylinder, resummed to all orders in the compact holonomy rather than truncated at leading order. The result proves that finite-circumference corrections to the topological response are strictly exponential, with no universal $1/L$ term, and fixes the precise map from microscopic spinon Chern number to physical spin Hall conductance. We validate this chain of reasoning on the kagome lattice at three independent levels: an exact parton band-structure calculation ($C=-1$, converging exponentially over cylinders four to twelve sites wide), and an interacting density-matrix renormalization group flux pump ($ν_s=-0.500\pm0.011$) that agrees with the analytic prediction without any adjustable parameter. Together, these results turn a one-loop anomaly calculation into a quantitatively verified bridge between microscopic topology and observable fractional response.