Figure 3

Analysis of the MI-SF boundary. Left panel: system-size dependence of the charge gap \(\Delta E_{c}\) per site, for fixed \(t/\Omega= 0.25\) and two values of \(g_{2}\) in the MI (\(g_{2}/\Omega= 1.35\)) and in the SF (\(g_{2}/\Omega= 1.5\)) phase. Symbols denote the DMRG results. Solid and dashed lines are linear and quadratic fitting curves, respectively. The difference between the extrapolated values in the two fits \(\Delta E_{c}^{\infty} = \lim_{L \to\infty} \Delta E_{c}^{(L)}\) is negligible. Right panel: determination of the critical \(g_{2}^{*}\) value for the quantum phase transition. The triangles denote the charge gap per site \(\Delta E_{c}^{\infty}\) at the thermodynamic limit, as extrapolated in the left panel. The dashed line is a best linear fit of the data vs. \(g_{2}\). The critical point is obtained when \(\Delta E_{c}\) vanishes. For \(t / \Omega=0.25\), we get \(g_{2}^{*} / \Omega\approx1.379\).