Single Dirac Fermions on a Lattice: A Unified View of Chiral and Parity Anomalies in Condensed Matter
Abstract
Quantum anomalies are usually formulated in continuum field theory, where regularization may violate a classical symmetry. In a crystal, the compact Brillouin zone and the high-energy bands make that ultraviolet completion physical. This Perspective develops a unified lattice view of chiral and parity anomalies through the problem of realizing a single massless Dirac fermion. The central motif is a finite low-energy region in which the relevant symmetry is restored, embedded in a high-energy sector that breaks the symmetry and completes the lattice theory. In 1+1 and 3+1 dimensions, the quantized chiral response is protected by local chiral symmetry near the Fermi surface. In 2+1 dimensions, the half-integer Hall response belongs to a single massless Dirac band with mirror symmetry, not to a generic massive Dirac insulator that breaks the symmetry explicitly. In both cases the lattice perspective reveals how the low-energy symmetry emerges from the high-energy sector and protects the anomalous and quantized response. Recent experimental progress in quantum materials is discussed in light of this unified view.
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BibTeX
@misc{shen2026single,
title = {{Single Dirac Fermions on a Lattice: A Unified View of Chiral and Parity Anomalies in Condensed Matter}},
author = {Shun-Qing Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06349},
eprint = {2609.06349},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06349},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Single Dirac Fermions on a Lattice: A Unified View of Chiral and Parity Anomalies in Condensed Matter
AU - Shun-Qing Shen
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06349
UR - https://arxiv.org/abs/2609.06349
AB - Quantum anomalies are usually formulated in continuum field theory, where regularization may violate a classical symmetry. In a crystal, the compact Brillouin zone and the high-energy bands make that ultraviolet completion physical. This Perspective develops a unified lattice view of chiral and parity anomalies through the problem of realizing a single massless Dirac fermion. The central motif is a finite low-energy region in which the relevant symmetry is restored, embedded in a high-energy sector that breaks the symmetry and completes the lattice theory. In 1+1 and 3+1 dimensions, the quantized chiral response is protected by local chiral symmetry near the Fermi surface. In 2+1 dimensions, the half-integer Hall response belongs to a single massless Dirac band with mirror symmetry, not to a generic massive Dirac insulator that breaks the symmetry explicitly. In both cases the lattice perspective reveals how the low-energy symmetry emerges from the high-energy sector and protects the anomalous and quantized response. Recent experimental progress in quantum materials is discussed in light of this unified view.
ER -