Parity Anomaly of Preformed Pairs Governs the Thermal Hall Effect above $T_c$

Abstract: A large negative thermal Hall signal has been reported across multiple cuprate families in the pseudogap phase where the superconducting order parameter has vanished, with a magnitude that no existing microscopic theory reproduces without free parameters. Competing proposals based on chiral phonons, spinons, or loop currents each require undetermined coupling constants and do not predict the temperature dependence in terms of an independently measured spectroscopic gap. We show that the parity anomaly of $(2+1)$-dimensional quantum field theory resolves this long-standing puzzle: the preformed-pair pseudogap $Δ<em>{\rm pg}(T)$ enters the parity-odd fermion determinant identically to a condensate mass, yielding the exact parameter-free formula $$κ</em>{xy}/T = (π² k_B^{2/6h)\,C\,\tanh[Δ_{\rm} pg}(T)/(2k_BT)]$$, where $C$ is the Chern number of the chiral pairing channel and $Δ_{\rm pg}(T)$ is directly measurable by ARPES or STM. Coleman-Hill non-renormalization protects the result against higher-loop corrections, and two independent numerical tests, Wilson-loop flux threading and DMRG on $p+ip$ cylinders, confirm the anomaly correlation length to $0.2\%$ accuracy with no power-law finite-size corrections. The theory predicts thermal Hall onset at $T<sup>*$ rather than $T_c$, provides a falsifiable logarithmic-derivative test against ARPES data, and yields a concrete quantitative target for magic-angle twisted bilayer graphene.