In the study of $\mathcal{P}\mathcal{T}$-symmetric quantum systems with non-Hermitian perturbations, one of the most important questions is whether eigenvalues stay real or whether $\mathcal{P}\mathcal{T}$ symmetry is spontaneously broken when eigenvalues meet. A particularly interesting set of eigenstates is provided by the degenerate ground-state subspace of systems with topological order. In this paper, we present simple criteria that guarantee the protection of $\mathcal{P}\mathcal{T}$ symmetry and, thus, the reality of the eigenvalues in topological many-body systems. We formulate these criteria in both geometric and algebraic form and demonstrate them using the toric code and several different fracton models as examples. Our analysis reveals that $\mathcal{P}\mathcal{T}$ symmetry is robust against a remarkably large class of non-Hermitian perturbations in these models; this is particularly striking in the case of fracton models due to the exponentially large number of degenerate states.

中文翻译：

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拓扑多体系统中奇偶时间对称性的保护：非赫米特复曲面代码和fracton模型

在研究中 $\mathcal{P}\mathcal{Ť}$非厄米特摄动的非对称量子系统，最重要的问题之一是特征值是否保持实数或 $\mathcal{P}\mathcal{Ť}$特征值相遇会自动破坏对称性。具有拓扑顺序的系统的退化基态子空间提供了一组特别有趣的本征态。在本文中，我们提出了一些简单的标准来确保对$\mathcal{P}\mathcal{Ť}$拓扑多体系统中特征值的对称性和真实性。我们以几何和代数形式制定这些标准，并以复曲面代码和几种不同的fracton模型为例进行说明。我们的分析表明$\mathcal{P}\mathcal{Ť}$在这些模型中，对称性对于非赫米特扰动类别非常强大；在fracton模型的情况下，这尤其明显，因为退化状态的数量呈指数级增长。