We apply a general technique, based on the similarity renormalization procedure introduced by Glazek and Wilson, to investigate the strong coupling expansion of the Bose-Hubbard model at half filling. This approach provides a mapping of the Bose-Hubbard model onto an XXZ spin-1/2 magnetic chain,  with renormalized exchange energy J and renormalized anisotropy parameter Delta.  We then exploit  the  analytical results available for the spin-1/2 XXZ model in a finite chain of length N to compute the correlation functions of the Bose-Hubbard model and to compare them with those obtained from the numerical density matrix renormalization group approach; the agreement is very good,  even at N = 30. Our approach allows to engineer a reliable quantum magnetic analog of the Bose-Hubbard model, enabling us to show that the superfluid-charge density wave (superfluid-Mott insulator)  phase transition of the Bose-Hubbard  model corresponds,  for Delta =1 (Delta =-1)  to the spin liquid-antiferromagnet (spin liquid-ferromagnet)  phase transition of the XXZ-model.

[with the collaboration of P. Sodano (Università di Perugia), D. Rossini (SNS-Pisa), A. Trombettoni (SISSA)]