Emergence of the Gibbs ensemble as a steady state in Lindbladian dynamics

We explicitly construct unique nonequilibrium steady state (NESS) of the Lindblad master equation characterized by a Gibbs ensemble ${\ensuremath{\rho}}_{\text{NESS}}\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\beta}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}}$, where the effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ consists only of $\text{U}(1)$ conserved charges of the original Hamiltonian. Specifically, when the original Hamiltonian has multiple charges, it is possible to couple them with bathes at different temperature, respectively, but still leads to an equilibrium state. To access the Gibbs NESS, the jump operators need to be properly chosen to fulfill quantum detailed balance condition (qDBC). These jump operators are ladder operators for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ and jump process they generate form a vertex-weighted directed acyclic graph (wDAG). By studying the $XX$ model and Fredkin model, we showcase how the Gibbs state emerges as an equilibrium steady state.