For parity-conserving fermionic chains, we review how to associate [Formula: see text]-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the [Formula: see text]-valued spectral flow provides a topological obstruction for two systems to have the same [Formula: see text]-index. A rudimentary definition of a [Formula: see text]-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

中文翻译：

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关于费米子链基态的 ℤ2-指数

对于宇称守恒的费米子链，我们回顾了如何将[公式：见正文]-索引与具有二次和高阶相互作用的有限系统中的基态以及无限 CAR 代数上的准自由基态相关联。结果表明，[公式：见文本]-值谱流为两个系统提供了拓扑障碍，使两个系统具有相同的 [公式：见文本]-索引。还提供了一维无限 CAR 代数的一类奇偶性不变和纯基态的 [公式：见文本]-相位标签的基本定义。如果不关闭无限 GNS 哈密顿量的光谱间隙，则无法连接具有不同相位标签的基态。