We use two recently proposed methods to calculate exactly the spectrum of two spin-${1over 2}$ charge carriers moving in a ferromagnetic background, at zero temperature, for three types of models. By comparing the low-energy states in both the one-carrier and the two-carrier sectors, we analyze whether complex models with multiple sublattices can be accurately described by simpler Hamiltonians, such as one-band models. We find that while this is possible in the one-particle sector, the magnon-mediated interactions which are key to properly describe the two-carrier states of the complex model are not reproduced by the simpler models. We argue that this is true not just for ferromagnetic, but also for antiferromagnetic backgrounds. Our results question the ability of simple one-band models to accurately describe the low-energy physics of cuprate layers.
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