From perfect to imperfect poor man's Majoranas in minimal Kitaev chains

Poor man's Majoranas (PMMs) hold the promise to engineer Majorana bound states in a highly tunable setup consisting of a chain of quantum dots that are connected via superconductors. Due to recent progress in controlling the amplitudes of elastic cotunneling (ECT) and crossed Andreev reflection (CAR), two vital ingredients for PMMs, experimental investigations of PMMs have gained significant interest. Previously, analytic conditions for the ``sweet spots'' that result in PMMs have focused on systems with infinite Zeeman energy. Here, we derive analytically a sweet spot condition for PMMs in a system with finite Zeeman energy in the absence of Coulomb interaction. We then consider two numerical models, one in which ECT and CAR are transmitted via superconducting bulk states and one in which they are transmitted via an Andreev bound state. We demonstrate that the analytical sweet spot conditions can only be approximated in these more realistic models, but they cannot be satisfied exactly. As a consequence, we do not find perfect PMMs in these systems, but instead near-zero-energy states that are highly, but not perfectly, localized. These states can be considered as imperfect PMMs and their classification relies on threshold values, which adds some arbitrariness to the concept of PMMs.