Parity anomaly governs the thermal Hall response of chiral superconductivity in rhombohedral graphene above and below $T_c$

RESUMO

Chiral superconductivity has recently been confirmed in rhombohedral tetra- and pentalayer graphene near the BCS--BEC crossover, but no theory tells the experimentalist what thermal Hall response to expect, or why any signal should persist above the phase-coherence temperature $T_c$. We show that the parity anomaly of $(2{+}1)$-dimensional field theory fixes the answer exactly, at all temperatures, with no free parameters: $\kappa_{xy}/T = (\pi^2 k_B^2/6h)\,C_{\rm BdG}\,\tanh[\Delta(T)/(2k_BT)]$, where $C_{\rm BdG}$ is the BdG Chern number and $\Delta(T)$ is the fermionic excitation gap. The BCS--BEC two-gap relation $\Delta^2 = \Delta_{\rm sc}^2 + \Delta_{\rm pg}^2$ makes the same formula govern both the condensate and pseudogap regimes, so the signal onsets at the pair-formation temperature $T^{*}$ rather than at $T_c$. Coleman--Hill non-renormalization and the $c_1 = 0$ theorem protect this result against interactions and finite-size artefacts. Three independent numerical validations confirm the topological input at machine precision (FHS Chern numbers, Wilson-loop $c_1 = 0$ test) and at the many-body level (DMRG on 28 converged ground states, including the real-space $p+ip$ signature $\arg\mathcal{A}_y - \arg\mathcal{A}_x = -\pi/2$ recovered to $10^{-14}$). The theory delivers an immediate falsifiable test that requires no new experimental apparatus: the sign of $\kappa_{xy}/T$ below $T_c$ must equal the sign of the anomalous Hall resistance $R_{xy}$ already measured above $T_c$, and four further predictions accessible by dilution-refrigerator nano-calorimetry on existing devices.