Dissipation meets conformal interface in open quantum systems: How the relaxation rate is suppressed

Conformal interfaces play an important role in quantum critical systems. In closed systems, the transmission properties of conformal interfaces are typically characterized by two quantities: One is the effective central charge $c_{\text{eff}}$, which measures the amount of quantum entanglement through the interface, and the other is the transmission coefficient $c_{\text{LR}}$, which measures the energy transmission through the interface. In the present work, to characterize the transmission property of conformal interfaces in open quantum systems, we propose a third quantity $c_{\text{relax}}$, which is defined through the ratio of Liouvillian gaps with and without an interface. Physically, $c_{\text{relax}}$ measures the suppression of the relaxation rate towards a steady state when the system is subject to a local dissipation. We perform both analytical perturbation calculations and exact numerical calculations based on a free fermion chain at the critical point. It is found that $c_{\text{relax}}$ decreases monotonically with the strength of the interface. In particular, $0\le c_{\text{relax}}\le c_{\text{LR}}\le c_{\text{eff}}$, where the equalities hold if and only if the interface is totally reflective or totally transmissive. Our result for $c_{\text{relax}}$ is universal in the sense that $c_{\text{relax}}$ is independent of (i) the dissipation strength in the weak dissipation regime and (ii) the location where the local dissipation is introduced. Comparing to the previously known $c_{\text{LR}}$ and $c_{\text{eff}}$ in a closed system, our $c_{\text{relax}}$ shows a distinct behavior as a function of the interface strength, suggesting its novelty to characterize conformal interfaces in open systems and offering insights into critical systems under dissipation.