Parity Anomaly as Modular Commutator with Massless Dirac Fermion
The modular commutator $J(A,B,C) = i\langle[K_{AB},K_{BC}]\rangle$ extracts the chiral central charge $c_-$ from a single bulk wavefunction of a \emph{gapped} 2d state, where $3J/\pi=c_-$. 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 \textit{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 \textit{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 $3J/\pi = c_-+\frac{1}{2}$. 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.