Parity Anomaly as Modular Commutator with Massless Dirac Fermion

The modular commutator $J(A,B,C)=i⟨[K_{AB},K_{BC}]⟩$ extracts the chiral central charge $c_{-}$ from a single bulk wavefunction of a \emph{gapped} 2d state, where $3J\mathrm{/}\pi =c_{-}$. Inspired by the recent developments in the field of gapless symmetry-protected topological phases, we ask: what does the modular commutator measure, if it is well-defined at all, when the 2d bulk becomes $\mathit{\text{gapless}}$? Several interesting new insights can already be obtained using the simple Haldane honeycomb model. For the critical point hosting an isolated Dirac node we find that $J$ remains sharp: it converges to a $\mathit{\text{half-quantized}}$ value, with corrections that decay as a power law in the subsystem size rather than exponentially, mirroring the power-law correlations in gapless systems. We prove the half-quantization using an emergent reflection symmetry of the massless Dirac cone, and show that the half-quantized contribution comes from the other gapped cone (the massive partner of the massless one). This massive partner can be interpreted as the physical incarnation of the Pauli-Villars regulator, which is the origin of the parity-breaking level-$\frac{1}{2}$ Chern-Simons term (with half-quantized Hall conductance) and the parity anomaly. When protected chiral edge modes coexist with a bulk Dirac node we obtain $3J\mathrm{/}\pi =c_{-}+\frac{1}{2}$. The half-quantization is also shown to be robust against tripartition deformation, tuning Dirac velocity and Dirac cone anisotropy. We further investigate other types of gaplessness—quadratic nodes (in contrast to linear Dirac) and the case with Fermi surface—and show that the robust half-quantization of $J$ is lost in such non-Dirac cases. These results generalize the modular commutator beyond gapped phases, and at the same time provide an information-theoretic measurement of the parity anomaly.