  Important properties of complex quantum many-body systems and their phase
diagrams can often already be inferred from the impurity limit. The Bose
polaron problem describing an impurity atom immersed in a Bose-Einstein
condensate is a paradigmatic example. One of the most interesting features of
this model is the competition between the emergent impurity-mediated attraction
between the bosons and their intrinsic repulsion. The arising higher-order
correlations make the physics rich and interesting, but also complex to
describe theoretically. To tackle this challenge, we develop a quantum
chemistry-inspired computational technique and compare two state-of-the-art
variational methods that fully include both the boson-impurity and boson-boson
interactions on a non-perturbative level. For a sweep of the boson-impurity
interaction strength, we find two regimes of qualitatively different behaviour.
If the impurity-mediated interactions overcome the repulsion between the
bosons, the polaron becomes unstable due to the formation of large bound
clusters. If instead the interboson interactions dominate, the impurity will
experience a crossover from a polaron into a small molecule. We achieve a
unified understanding incorporating both of these regimes and the transition
between them. We show that both the instability and crossover regime can be
studied in realistic cold-atom experiments. Moreover, we develop a simple
analytical model that allows us to interpret these phenomena in the typical
Landau framework of first-order phase transitions that turn second-order at a
critical endpoint, revealing a deep connection of the Bose polaron model to
both few- and many-body physics.
