AI 中文总结
该研究将模对易子推广至无隙二维系统，发现含孤立狄拉克节点时其收敛至半量子化值，为宇称反常提供信息论测量，且半量子化对多种扰动鲁棒。
AI 中文摘要
模对易子 $J(A,B,C)=i\langle[K_{AB},K_{BC}]\rangle$ 可从带隙二维态的单个体波函数中提取手性中心电荷 $c_-$，满足 $3J/\pi=c_-$。受无对称性保护拓扑相领域最新进展启发，我们提出问题：若二维体态变为无隙态，模对易子（若其有明确定义）会测量到什么？利用简单的Haldane蜂窝模型已能获得若干有趣新见解。对于含孤立狄拉克节点的临界点，我们发现 $J$ 仍具有明确值：它收敛至半量子化值，修正项随子系统大小呈幂律衰减而非指数衰减，这与无隙系统的幂律关联一致。我们利用无质量狄拉克锥的涌现反射对称性证明了半量子化，并表明半量子化贡献来自另一个带隙锥（即无质量锥的有质量伙伴）。该有质量伙伴可被解释为Pauli-Villars正则化器的物理体现，这是宇称破缺的水平-1/2陈-西蒙斯项（含半量子化霍尔电导）及宇称反常的起源。当受保护的手性边缘模式与体态狄拉克节点共存时，我们得到 $3J/\pi=c_-+1/2$。还证明了半量子化对三分体变形、狄拉克速度及狄拉克锥各向异性具有鲁棒性。我们进一步研究其他类型的无隙性——二次节点（与线性狄拉克节点相对）及费米面情形，发现在这类非狄拉克情形中，$J$ 的鲁棒半量子化消失。这些结果将模对易子的适用范围扩展至带隙相之外，同时为宇称反常提供了一种信息论测量方法。
英文摘要
The modular commutator $J(A,B,C) = i\langle[K_{AB},K_{BC}]\rangle$ extracts the chiral central charge $c_-$ from a single bulk wavefunction of a \emph{gapped} 2d state, where $3J/π=c_-$. Inspired by the recent developments in the field of gapless symmetry-protected topological phases, we ask: what does the modular commutator measure, if it is well-defined at all, when the 2d bulk becomes \textit{gapless}? Several interesting new insights can already be obtained using the simple Haldane honeycomb model. For the critical point hosting an isolated Dirac node we find that $J$ remains sharp: it converges to a \textit{half-quantized} value, with corrections that decay as a power law in the subsystem size rather than exponentially, mirroring the power-law correlations in gapless systems. We prove the half-quantization using an emergent reflection symmetry of the massless Dirac cone, and show that the half-quantized contribution comes from the other gapped cone (the massive partner of the massless one). This massive partner can be interpreted as the physical incarnation of the Pauli-Villars regulator, which is the origin of the parity-breaking level-$\frac{1}{2}$ Chern-Simons term (with half-quantized Hall conductance) and the parity anomaly. When protected chiral edge modes coexist with a bulk Dirac node we obtain $3J/π= c_-+\frac{1}{2}$. The half-quantization is also shown to be robust against tripartition deformation, tuning Dirac velocity and Dirac cone anisotropy. We further investigate other types of gaplessness---quadratic nodes (in contrast to linear Dirac) and the case with Fermi surface---and show that the robust half-quantization of $J$ is lost in such non-Dirac cases. These results generalize the modular commutator beyond gapped phases, and at the same time provide an information-theoretic measurement of the parity anomaly.
Comments5 pages with 4 figures in main text + 2 pages with 4 figures in appendix
