  The Bose Hubbard model (BHM) is an archetypal quantum lattice system
exhibiting a quantum phase transition between its superfluid (SF) and
Mott-insulator (MI) phase. Unlike in higher dimensions the phase diagram of the
BHM in one dimension possesses regions in which increasing the hopping
amplitude can result in a transition from MI to SF and then back to a MI. This
type of re-entrance is well known in classical systems like liquid crystals yet
its origin in quantum systems is still not well understood. Moreover, this
unusual re-entrant character of the BHM is not easily captured in approximate
analytical or numerical calculations. Here we study in detail the predictions
of three different and widely used approximations; a multi-site mean-field
decoupling, a finite-sized cluster calculation, and a real-space
renormalization group (RG) approach. It is found that mean-field calculations
do not reproduce re-entrance while finite-sized clusters display a precursor to
re-entrance. Here we show for the first time that RG does capture the
re-entrant feature and constitutes one of the simplest approximation able to do
so. The differing abilities of these approaches reveals the importance of
describing short-ranged correlations relevant to the kinetic energy of a MI in
a particle-number symmetric way.
