Bose-Fermi-Hubbard model in the truncated Hilbert space limit

It is shown that in the Bose-Fermi-Hubbard model which is used for a description of the ultracold atomic boson-fermion mixture in the optical lattice, the $n_\text{B}\leqslant 2$ restriction enables one to analyze a more general case of separated lobes Mott insulator in comparison with the case of $n_\text{B} \leqslant 1$ (hard-core bosons). It also showed that the restriction to no more than 2 bosons on site is enough to comprehend the structure of $(\mu, |t_0|)$ diagrams at arbitrary $n^\text{b}$ values with an account of a possibility of the 1st order phase transition.