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Symmetries in an open quantum system lead to degenerated Liouvillian thatphysically implies the existence of multiple steady states. In such cases,obtaining the initial condition independent stead states is highly nontrivialsince any linear combination of the \emph{true} asymptotic states, which maynot necessarily be a density matrix, is also a valid asymptote for theLiouvillian. Thus, in this work we consider different approaches to obtain the\emph{true} steady states of a degenerated Liouvillian. In the ideal scenario,when the open system symmetry operators are known we show how these can be usedto obtain the invariant subspaces of the Liouvillian and hence the steadystates. We then discuss two other approaches that do not require any knowledgeof the symmetry operators. These could be a powerful tool to deal with quantummany-body complex open systems. The first approach which is based onGramm-Schmidt orthonormalization of density matrices allows us to obtain\emph{all} the steady states, whereas the second one based on large deviationsallows us to obtain the non-degenerated maximum and minimum current-carryingstates. We discuss our method with the help of an open para-Benzene ring andexamine interesting scenarios such as the dynamical restoration of Hamiltoniansymmetries in the long-time limit and apply the method to study theeigenspacing statistics of the nonequilibrium steady state.

Degenerated Liouvillians and Steady-State Reduced Density Matrices