Single-site entanglement at the superconductor-insulator transition in the Hirsch model

We investigate the transition to the insulating state in the one-dimensional Hubbard model with bond-charge interaction x (Hirsch model), at half-filling and $T=0$. By means of the density-matrix renormalization group algorithm the charge gap closure is examined by both standard finite-size scaling analysis and looking at singularities in the derivatives of single-site entanglement. The results of the two techniques show that a quantum phase transition takes place at a finite Coulomb interaction uc(x) for x0.5. The region 0<u<uc turns out to have a superconducting nature, at least for not too large x>xc.