Parity-Anomalous Hall Response of Chiral Lifshitz Fermions

The parity anomaly of a single massless Dirac fermion is manifested as a half-quantized Hall response once the required parity-breaking regularization is supplied. We study the corresponding response for two-dimensional chiral Lifshitz fermions. For a node with low-energy coupling (qx +iqy) N, the Fermi-loop winding number W = N gives an anomalous Hall response σxy anom = (W/2)e 2/h. We examine this mechanism in chiral multilayer Haldane models tuned to a valley-selective critical line, where a single gapless chiral Lifshitz fermion remains at the K valley. For the AB, ABC, and ABCA stacks, the projected pseudospin textures give W = 2, 3, 4, while Kubo calculations yield σxy = (1, 3/2, 2)e 2/h; integrated equilibrium boundary currents reproduce the same W/2 sequence. Six-terminal simulations with dephasing further show that the Hall conductivity is quantized at W/2 while the longitudinal conductivity remains finite. These results identify gapless chiral Lifshitz fermions as a higher-order, winding-controlled generalization of the parity-anomalous Hall response, distinct from a quantum anomalous Hall insulating state even when W/2 is an integer.