Logarithmic entanglement scaling in dissipative free-fermion systems

We study the quantum information spreading in one-dimensional free-fermion systems in the presence of localized thermal baths. We employ a nonlocal Lindblad master equation to describe the system-bath interaction, in the sense that the Lindblad operators are written in terms of the Bogoliubov operators of the closed system, and hence are nonlocal in space. The statistical ensemble describing the steady state is written in terms of a convex combination of the Fermi-Dirac distributions of the baths. Due to the singularity of the free-fermion dispersion, the steady-state mutual information exhibits singularities as a function of the system parameters. While the mutual information generically satisfies an area law, at the singular points it exhibits logarithmic scaling as a function of subsystem size. By employing the Fisher-Hartwig theorem, we derive the prefactor of the logarithmic scaling, which depends on the parameters of the baths and plays the role of an effective “central charge.” This is upper bounded by the central charge governing ground-state entanglement scaling. We provide numerical checks of our results in the paradigmatic tight-binding chain and the Kitaev chain.