A mathematical quantity built from overlapping regions of a quantum state approached one-half in a model with a single Dirac cone, according to a preprint. The quantity, called a modular commutator, compares the modular Hamiltonians associated with those regions; each Hamiltonian is obtained from the region's reduced density matrix. The scaled modular commutator moved toward one-half as the system grew, with finite-size corrections that fell as a power law.
The same single-Dirac gapless bulk retained a leading entanglement area law, an arrangement in which the dominant scaling is tied to the boundary of the chosen region, although its area-law structure differed from the gapped formulation. That distinction suggests the half-unit is not simply a feature of any system without a gap: the quadratic-node and Fermi-surface examples showed different modular-commutator behavior.
The calculation behind the signal
The calculation stayed within free-fermion ground states from variants of the Haldane honeycomb model. Its cases included an original Haldane system at a single Dirac-node transition; a coupled double-layer system with a chiral edge mode and a bulk Dirac node; a modified-hopping quadratic node; and a doped Fermi surface. Restricted two-point correlation matrices determined the Gaussian reduced states and single-particle modular Hamiltonians used in the free-fermion calculations.
What the half-unit may mean
The authors interpret the half-quantized contribution as an entanglement manifestation of the parity anomaly. In their account, a massive partner cone is the parity-breaking contribution and is viewed as a physical Pauli-Villars regulator for the massless Dirac fermion. That is the authors' mechanism-level interpretation of the model result.
Within the tested variations, the half-quantization held as the tripartition was deformed and as the Dirac velocity and cone anisotropy were changed. The double-layer example placed a chiral edge mode alongside the bulk Dirac node. The modular commutator then converged to the average of the adjacent gapped phases and included both bulk and edge contributions, which the authors describe as measuring total chirality.
Other gapless cases tell a different story
The quadratic-node case marked a clear break from that pattern. It still obeyed an area law, but its modular commutator differed significantly from the average of the adjacent gapped phases. Tests found that the value was not strongly affected by the tripartition geometries used, yet it did depend on the curvature of the node.
The doped Fermi-surface case was less orderly still. Its entanglement entropy fit the Widom form, scaling as R ln R + O(R), while its modular commutator was not half-quantized, changed with tripartition deformation and was described as not well-defined. The contrast leaves the Dirac result distinct from these other forms of gaplessness.
The study also checked conditional mutual information, an information measure for how regions remain linked. At the Dirac critical point, it saturated as subsystem sizes increased instead of decaying exponentially as in the gapped case, indicating a violation of the Markov property. A separate single-wavefunction Hall-conductance estimator was also half-quantized at the Dirac critical point and showed power-law convergence.
A result with a narrow reach
The result should be read as a finding about the free-fermion models studied, not as a statement about gaplessness in general. All reported calculations were for free fermions, and interacting gapless systems were not tested. The separate Hall-conductance result was an information-theoretic estimator within the model, not a direct experimental transport measurement.
The manuscript is labeled arXiv:2608.26078v1. The work was supported in part by the Alexander von Humboldt Foundation through a Humboldt Research Fellowship.
Paper data and sources
Original title: Parity Anomaly as Modular Commutator with Massless Dirac Fermion
Authors: Meng Zeng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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