Topological classifications of quadratic bosonic excitations in closed and open systems with examples

The topological classifications of quadratic bosonic systems according to the symmetries of the dynamic matrices from the equations of motion of closed systems and the effective Hamiltonians from the Lindblad equations of open systems are analyzed. While the non-Hermitian dynamic matrix and effective Hamiltonian both lead to a ten-fold way table, the system-reservoir coupling may cause a system with or without coupling to a reservoir to fall into different classes. A 2D Chern insulator is shown to be insensitive to the different classifications. In contrast, we present a 1D bosonic Su-Schrieffer-Heeger model with chiral symmetry and a 2D bosonic topological insulator with time-reversal symmetry to show the corresponding open systems may fall into different classes if the Lindblad operators break the symmetry.