Fundamental pathologies in Lindblad descriptions of systems weakly coupled to baths

It is very common in the literature to write down a Lindblad equation to describe a system with multiple degrees of freedom weakly connected to multiple thermal baths which can, in general, be at different temperatures and chemical potentials. However, the microscopically derived quantum master equation up to leading order in system-bath coupling is of so-called Redfield form which is known to not preserve complete positivity in most cases. Here, we analytically show in generality that, in such cases, enforcing complete positivity by imposing any Lindblad form, via any further approximation, necessarily leads to either violation of thermalization, or violation of local conservation laws in non-equilibrium steady state (NESS) due to inaccurate coherences in energy eigenbasis. In other words, a weak system-bath coupling quantum master equation that is completely positive, shows thermalization and preserves local conservation laws in NESS is fundamentally impossible in generic situations. On the other hand, the Redfield equation, although generically not completely positive, shows thermalization, always preserves local conservation laws and gives correct coherences to leading order. We exemplify our analytical results numerically in an interacting open quantum spin system.