Linear Response Theory of Fermionic Lindbladian Systems

For sufficiently weak external perturbation, the response of a system to that perturbation is usually linear in the perturbation. In closed quantum systems this can be described by the Kubo formalism, but in open systems where energy and particles enter and leave the system the Kubo formalism breaks down. Here we develop a linear response formalism for open quantum systems comprised of fermions governed by the Lindblad master equation. The Lindblad master equation describes the time evolution of a system which captures the quantum processes of decoherence and collapse; these processes are fundamentally different from the classical non-Hermitian processes of gain and loss. Using this formalism we present general expressions that can be used to calculate the optical conductivity and DC electromagnetic response of open fermionic systems. This builds on our previous work where we developed a "single-particle Lindbladian" approach to studying open fermionic systems and derived a flat-band condition under which flat bands form in a broad class of systems.