The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U(1) symmetry and the DIII class without U(1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.

中文翻译：

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高斯态中保对称二次 Lindbladian 和耗散驱动的拓扑转变


开放量子系统的动态演化可以由密度矩阵的 Lindblad 方程控制。在本文中，我们建议通过其模哈密顿量的拓扑不变量来表征密度矩阵拓扑。由于此类哈密顿量的拓扑分类取决于其对称类，因此我们解决的一个主要问题是确定 Lindbladian 算子的要求，在该要求下，模块化哈密顿量可以在动态演化过程中保留其对称类。我们针对费米子高斯态和作为一组费米子算子的二次算子的模哈密顿量来解决这个问题。当满足这些条件时，再加上模哈密顿量的对称类的非平凡拓扑分类，随着时间的演变就会发生拓扑转变。我们提出了耗散驱动拓扑转变的两个例子，其中模哈密顿量位于具有 U(1) 对称性的 AIII 类和不具有 U(1) 对称性的 DIII 类中。通过有限尺寸缩放，我们表明这种密度矩阵拓扑转变发生在有限时间。我们还展示了这一转变的物理特征。