Dissipation Dynamics Driven Transitions of the Density Matrix Topology

The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this letter, we propose that the density matrix topology can undergo a transition during the Lindbladian dynamical evolution. Here we characterize the density matrix topology by the topological invariant of its modular Hamiltonian. We focus on the fermionic Gaussian state, where the modular Hamiltonian is a quadratic operator of a set of fermionic operators. The topological classification of such Hamiltonians depends on their symmetry classes. Hence, a primary issue we deal with in this work is to determine the requirement for the Lindbladian operators, under which the modular Hamiltonian can maintain its symmetry class during the dynamical evolution. 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 in the DIII class without U(1) symmetry, respectively. As a manifestation of the topological transition, we present the signature of the eigenvalues of the density matrix at the transition point.