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| Format: | Recurso digital |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19622631 |
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Table of Contents:
- <p>The Nielsen-Ninomiya theorem dictates that any local, translationally invariant, and Hermitian discrete Dirac operator on a continuous infinite-volume Brillouin zone must produce fermion doublers. In this paper, we construct a discrete Hermitian Dirac operator, D_SSM(k), based on the 12 unit direction vectors of the Face-Centered Cubic (FCC) root system. We explicitly distinguish this topological 12-direction operator from the standard discrete Dirac operator on a physical FCC crystal lattice. We demonstrate a novel geometric evasion mechanism: number-theoretic sequestering. While the operator satisfies the Nielsen-Ninomiya theorem on the continuous torus T^3 (producing 8 interior doublers and additional boundary zero-modes), these zeros are located at strictly irrational fractional momentum coordinates (f_i in {0, ± 1/(2√2)}). Geometrically, this manifests as an incommensurate cuboctahedral network—a topological Moiré pattern where the local connectivity is irrationally out-of-phase with the global periodic boundaries. Because any finite periodic lattice of size L strictly restricts allowable momenta to rational fractions n/L, the entire continuous-limit doubler structure is kinematically inaccessible to any realizable finite system. We demonstrate via a dense 64^3 Brillouin-zone scan that the finite lattice spectrum contains exactly one massless mode at the Gamma-point. We prove analytically and verify numerically that the anticommutator {gamma_5, D_SSM(k)} vanishes identically at all momenta, preserving exact chiral symmetry without relying on explicit symmetry-breaking terms or non-Hermiticity, and verify that the spectral gap persists under a U(1) background gauge field.</p>