发表机构
Instituut-Lorentz, Universiteit Leiden（莱顿大学洛伦兹研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出采用切向色散关系的离散化方案，通过Cayley变换消除离散时间狄拉克模型的真空不稳定性，同时保留单个狄拉克锥，其施温格效应与连续介质结果一致。
AI 中文摘要
狄拉克量子行走（狄拉克方程的1+1维时空离散化）在准能量-动量布里渊区的中心和角落处均存在2μ的质量间隙。Gupta和Short最近指出[Quantum 9, 1845 (2025)]，狄拉克真空可向环境释放2μ的能量，在区角落处产生粒子-空穴对；他们以出现第二个低能狄拉克锥（费米子加倍）为代价消除了该真空不稳定性。本文表明，采用切向而非正弦色散关系的替代离散化方案，可在保留单个狄拉克锥的同时实现稳定性。关键步骤是将弗洛凯本征值的单位圆e^{-iε}通过Cayley变换映射到无界能量的实轴E=2tan(ε/2)。我们计算了切向费米子的施温格效应（均匀电场中的粒子-空穴对产生），结果表明其对产生率与单个狄拉克锥的连续介质结果一致。若标量势通过量子映射的厄米生成元耦合，区角落会完全解耦；若采用分裂算子耦合以保持晶格上的精确规范不变性，则区角落会有贡献，其权重随晶格常数二次方消失，这与狄拉克量子行走中连续介质极限下仍存在区角落不稳定性的情况形成对比。
英文摘要
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 $e^{-i\varepsilon}$ to the real line of unbounded energies $E=2\tan(\varepsilon/2)$. 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.
Comments11 pages, 6 figures
