非线性频率-动量拓扑与多重例外点的倍增
Nonlinear Frequency-Momentum Topology and Doubling of Multifold Exceptional Points
AI总结：
研究提出频率-动量 winding 数作为非线性多重例外点的拓扑不变量，推导出无对称性和有对称性条件下的倍增定理，揭示PT对称例外点的Z拓扑特性。
摘要：
即使在线性极限下，多重（也称为高阶）例外点在布里渊区内的拓扑缺乏一般描述，使倍增定理本质上仅限于二重例外点。本文建立了通过特征值引入非线性的n重例外点的倍增定理[EPn（n=2,3,…）]。为此，我们引入了新的拓扑不变量，称为频率-动量 winding 数，这些不变量在任意n和m（m≥n）的m带系统中描述整个布里渊区内的非线性EPn。这些不变量在无对称性和几种对称约束下（包括时间反演PT和电荷共轭- parity对称性）提供了统一的倍增定理证明。此外，即使在线性极限下，频率-动量 winding 数表明PT对称EP2的Z拓扑，超越了之前报告的Z2拓扑。频率-动量 winding 数还可扩展到一类通过本征向量引入非线性的耦合谐振器，其中频谱由频率的非线性标量方程决定。
英文摘要：
Even in the linear limit, the topology of multifold (also called higher-order) exceptional points across the Brillouin zone has lacked a general characterization, leaving the doubling theorem essentially limited to two-fold exceptional points. Here, we establish the doubling theorem of $n$-fold exceptional points [EP$n$s ($n=2,3,\ldots$)] for systems where nonlinearity enters through eigenvalues. To this end, we introduce new topological invariants, termed frequency-momentum winding numbers, which characterize nonlinear EP$n$s in $m$-band systems throughout the Brillouin zone for arbitrary $n$ and $m$ ($m\geq n$). These invariants enable a unified proof of the doubling theorem in the absence of symmetry and under several symmetry constraints, including parity-time ($PT$) and charge-conjugation-parity symmetries. Furthermore, even in the linear limit, the frequency-momentum winding number indicates $\mathbb{Z}$ topology of $PT$-symmetric EP$2$s which is beyond the previously reported $\mathbb{Z}_2$ topology. The frequency-momentum winding numbers can also be extended to a class of coupled resonators in which nonlinearity enters via the eigenvectors, whereas the spectrum is determined by a nonlinear scalar equation for the frequency.
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