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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.

Capturing the re‐entrant behavior of one‐dimensional Bose–Hubbard model