We present a full symmetry classification of fermion matter in and out of thermal equilibrium. Our approach starts from first principles, the ten different classes of linear and antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix. The object of classification is then the generators of reversible dynamics, dissipation and fluctuations, featuring’ in the generally irreversible and interacting dynamical equations. A sharp distinction between the symmetries of equilibrium and out-of-equilibrium dynamics, respectively, arises from the different role played by “time” in these two cases: In unitary quantum mechanics as well as in “microreversible” thermal equilibrium, antilinear transformations combined with an inversion of time define time-reversal symmetry. However, out of equilibrium an inversion of time becomes meaningless, while antilinear transformations in Fock space remain physically significant, and hence must be considered in autonomy. The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in nonequilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium. At large times, the dynamical generators thus symmetry classified determine the steady-state nonequilibrium distributions for arbitrary interacting systems. To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions. More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states. This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.

中文翻译：

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开放费米子量子物质的对称性类

我们提出了热平衡内外的费米子物质的完全对称分类。我们的方法从第一性原理，费米子Fock空间中的线性和反线性状态转换的十种不同类以及通过密度矩阵动力学方程的不变性定义的对称性出发。然后，分类的对象是可逆动力学，耗散和波动的生成器，其特征是通常不可逆且相互作用的动力学方程式。在这两种情况下，“时间”扮演的不同角色分别导致了平衡对称性和非平衡动力学之间的明显区别：在单一量子力学中以及在“微可逆”热平衡中，反变换与时间倒转相结合定义了时间反转对称性。但是，在不平衡状态下，时间倒置变得毫无意义，而Fock空间中的线性变换在物理上仍然很重要，因此必须在自治中予以考虑。这种二分法的实际结果是在非平衡量子动力​​学中实现了线性对称性（十个基本类别中的六个）的新实现，该对称性与已建立的热平衡规则根本不同。在很大程度上，如此对称地分类的动力发生器确定了任意相互作用系统的稳态非平衡分布。为了说明这一原理，我们考虑了相互作用的晶格费米子系统中对称保护拓扑相的固定。更普遍，我们考虑了以高斯态为代表的平均场相互作用系统的实用上非常重要的一类。此类自然以非Hermitian矩阵的语言描述，这使我们可以与文献中先前的分类方案进行比较。