Renormalization in the Theory of Open Quantum Systems via the Self-Consistency Condition

We investigate the topic of renormalization in the theory of weakly interacting open quantum systems. Our starting point is an open quantum system interacting with a single heat bath. For a given setup, we discuss that the stationary state of the Davies-GKSL equation is thermodynamically inconsistent with the presence of the Lamb-Stark shift term. For this reason, we postulate the self-consistency condition for the dynamical equations. The condition fixes the renormalization procedure and recovers the thermodynamical consistency. In this way, we rederive the cumulant equation to illustrate how the self-consistency condition enters the derivation of the dynamical equations. The physical interpretation of the renormalization procedure is discussed in terms of the Born approximation. Furthermore, we compare the Lamb-Stark shift term (dynamical correction) with the second-order (static) correction to the so-called mean-force (Gibbs state) Hamiltonian. The discrepancy between the static and the dynamical correction questions the physical meaning of the latter one. Finally, we formulate a simplified renormalization scheme that can be directly applied to Davies-GKSL or Bloch-Redfield equations.