Title: Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond Show Chinese title

Title: 多体林布兰德算子的对称分类：十重方式及其他

Abstract: We perform a systematic symmetry classification of many-body Lindblad superoperators describing general (interacting) open quantum systems coupled to a Markovian environment. Our classification is based on the behavior of the many-body Lindbladian under antiunitary symmetries and unitary involutions. We find that Hermiticity preservation reduces the number of symmetry classes, while trace preservation and complete positivity do not, and that the set of admissible classes depends on the presence of additional unitary symmetries: in their absence or in symmetry sectors containing steady states, many-body Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional symmetries and we consider non-steady-state sectors, they belong to a different set of 19 classes. In both cases, it does not include classes with Kramer's degeneracy. Remarkably, our classification admits a straightforward generalization to the case of non-Markovian, and even non-trace-preserving, open quantum dynamics. While the abstract classification is completely general, we then apply it to general (long-range, interacting, spatially inhomogeneous) spin-$1/2$ chains. We explicitly build examples in all ten classes of Lindbladians in steady-state sectors, describing standard physical processes such as dephasing, spin injection and absorption, and incoherent hopping, thus illustrating the relevance of our classification for practical physics applications. Finally, we show that the examples in each class display unique random-matrix correlations. To fully resolve all symmetries, we employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs related by symmetry operations. We further find that statistics of levels constrained onto the real and imaginary axes or close to the origin are not universal due to spontaneous breaking of Lindbladian PT symmetry. Show Chinese abstract

Abstract: 我们对描述一般（相互作用的）与马尔可夫环境耦合的开放量子系统的许多体林德布拉德超算符进行了系统性的对称分类。 我们的分类基于许多体林德布兰德算符在反酉对称性和酉对合下的行为。 我们发现，厄米性保持会减少对称类的数量，而迹保持和完全正性不会，且可接受的类集取决于是否存在额外的酉对称性：在它们缺失或包含稳态的对称性部分中，许多体林德布兰德算符属于十个非厄米对称类；然而，如果有额外的对称性并且考虑非稳态部分，则它们属于另一组19个类。 在这两种情况下，它都不包括克雷默退化类。 值得注意的是，我们的分类可以简单地推广到非马尔可夫情况，甚至是非迹保持的情况下的开放量子动力学。 虽然抽象分类是完全通用的，但我们随后将其应用于一般（长程、相互作用、空间不均匀）自旋-$1/2$链。 我们在所有十个稳态部分的林德布兰德算符类中明确构建了示例，这些示例描述了标准物理过程，如去相干、自旋注入和吸收以及非相干跳跃，从而展示了我们分类对于实际物理应用的相关性。 最后，我们表明每个类中的示例都显示出独特的随机矩阵相关性。 为了完全解析所有对称性，我们采用了复间隔比和由对称操作相关的本征向量对重叠的联合分析。 我们进一步发现，由于林德布兰德PT对称性的自发破缺，限制在实轴、虚轴或接近原点的能级统计不是普适的。