如何在格点上制定Majorana费米子的$\mathbb{Z}_8$拓扑不变量
How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice
AI总结：
本研究提出了一种在格点上计算Majorana费米子$\mathbb{Z}_8$拓扑不变量的方法，并通过数值验证展示了其与连续理论的一致性。
摘要：
拓扑不变量及其相关的反常在理解量子场论中的低能现象中发挥了关键作用。在格点规范理论中，标准的$\mathbb{Z}$-值的Atiyah-Singer索引是通过重叠Dirac算符通过Ginsparg-Wilson关系来制定的，但扩展到更一般的拓扑不变量仍有局限。在本工作中，我们提出了一种格点的Arf-Brown-Kervaire（ABK）不变量的制定方法，其值在$\mathbb{Z}_8$中。ABK不变量出现在具有反射对称性的二维非定向流形上的Majorana费米子配分函数中，其定义涉及对Dirac本征值的无限求和，必须进行适当的正则化。通过仔细处理边界条件，有无域壁质量项，我们证明ABK不变量可以从Wilson Dirac算符的Pfaffian中提取。我们进一步在二维格点上提供了数值验证，显示在环面、克莱因瓶、实射影平面和莫比乌斯带上的$\mathbb{Z}_8$值结果与连续理论中的结果一致。
英文摘要：
Topological invariants and their associated anomalies have played a crucial role in understanding low-energy phenomena in quantum field theories. In lattice gauge theory, the standard $\mathbb{Z}$-valued Atiyah-Singer index is formulated via the overlap Dirac operator through the Ginsparg-Wilson relation, but extensions to more general topological invariants have remained limited. In this work, we propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant, which takes values in $\mathbb{Z}_8$. The ABK invariant arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-oriented manifolds, and its definition involves an infinite sum over Dirac eigenvalues that must be properly regularized. By carefully treating the boundary conditions, with and without a domain-wall mass term, we demonstrate that the ABK invariant can be extracted from Pfaffians of the Wilson Dirac operator. We further provide numerical verification on two-dimensional lattices, showing that the $\mathbb{Z}_8$-valued results on the torus, Klein bottle, real projective plane, and Möbius strip agree with those in the continuum theory.
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