Optimized auxiliary representation of non-Markovian impurity problems by a Lindblad equation

We present a general scheme to address correlated nonequilibrium quantum impurity problems
based on a mapping onto an auxiliary open quantum system of small size. The infinite fermionic
reservoirs of the original system are thereby replaced by a small number NB of noninteracting auxiliary
bath sites whose dynamics are described by a Lindblad equation, which can then be exactly solved by
numerical methods such as Lanczos or matrix-product states. The mapping becomes exponentially
exact with increasing NB, and is already quite accurate for small NB. Due to the presence of the
intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. While in
previous work we put the focus on the manybody solution of the associated Lindblad problem, here
we discuss the mapping scheme itself, which is an essential part of the overall approach. On the one
hand, we provide technical details together with an in-depth discussion of the employed algorithms,
and on the other hand, we present a detailed convergence study. The latter clearly demonstrates the
above-mentioned exponential convergence of the procedure with increasing NB. Furthermore, the
influence of temperature and an external bias voltage on the reservoirs is investigated. The knowledge
of the particular convergence behavior is of great value to assess the applicability of the scheme to
certain physical situations. Moreover, we study different geometries for the auxiliary system. On the
one hand, this is of importance for advanced manybody solution techniques such as matrix product
states which work well for short-ranged couplings, and on the other hand, it allows us to gain more
insights into the underlying mechanisms when mapping non-Markovian reservoirs onto Lindblad-
type impurity problems. Finally, we present results for the spectral function of the Anderson impurity
model in and out of equilibrium and discuss the accuracy obtained with the different geometries of the
auxiliary system. In particular, we show that allowing for complex Lindblad couplings produces a
drastic improvement in the description of the Kondo resonance.