Tangent fermions can restore vacuum stability of discrete-time Dirac models
Abstract
The Dirac quantum walk (a 1+1 dimensional space-time discretization of the Dirac equation) has a 2μ mass gap both at the center and at the corner of the quasi-energy-momentum Brillouin zone. Gupta and Short recently noted [Quantum 9, 1845 (2025)] that the Dirac vacuum can create a particle-hole pair at the zone corner with the release of an energy 2μ to the environment. They removed this vacuum instability at the expense of fermion doubling, the appearance of a second low-energy Dirac cone. Here we show that an alternative discretization scheme, with a tangent rather than a sine dispersion relation, offers stability while retaining a single Dirac cone. The key step is the Cayley transformation from the unit circle of Floquet eigenvalues e−iε to the real line of unbounded energies E=2tan(ε/2). We compute the Schwinger effect (particle-hole pair creation in a uniform electric field) for tangent fermions and show that the pair-production rate agrees with the continuum result for a single Dirac cone. The zone corner is exactly decoupled if the scalar potential is coupled through the Hermitian generator of the quantum map. If it is coupled as a split operator, in order to preserve exact gauge invariance on the lattice, the corner does contribute - with a weight that vanishes quadratically with the lattice constants, in contrast to the Dirac quantum walk where the zone-corner instability survives the continuum limit.
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BibTeX
@misc{beenakker2026tangent,
title = {{Tangent fermions can restore vacuum stability of discrete-time Dirac models}},
author = {C. W. J. Beenakker},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01029},
eprint = {2609.01029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01029},
}
RIS
TY - UNPB
TI - Tangent fermions can restore vacuum stability of discrete-time Dirac models
AU - C. W. J. Beenakker
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01029
UR - https://arxiv.org/abs/2609.01029
AB - The Dirac quantum walk (a 1+1 dimensional space-time discretization of the Dirac equation) has a 2μ mass gap both at the center and at the corner of the quasi-energy-momentum Brillouin zone. Gupta and Short recently noted [Quantum 9, 1845 (2025)] that the Dirac vacuum can create a particle-hole pair at the zone corner with the release of an energy 2μ to the environment. They removed this vacuum instability at the expense of fermion doubling, the appearance of a second low-energy Dirac cone. Here we show that an alternative discretization scheme, with a tangent rather than a sine dispersion relation, offers stability while retaining a single Dirac cone. The key step is the Cayley transformation from the unit circle of Floquet eigenvalues e^−iε to the real line of unbounded energies E=2tan(ε/2). We compute the Schwinger effect (particle-hole pair creation in a uniform electric field) for tangent fermions and show that the pair-production rate agrees with the continuum result for a single Dirac cone. The zone corner is exactly decoupled if the scalar potential is coupled through the Hermitian generator of the quantum map. If it is coupled as a split operator, in order to preserve exact gauge invariance on the lattice, the corner does contribute - with a weight that vanishes quadratically with the lattice constants, in contrast to the Dirac quantum walk where the zone-corner instability survives the continuum limit.
ER -