Figure 4

Determination of the DW boundaries. The two-point correlation function \(C_{\mathrm{DW}}(r)\) for the polariton number and its asymptotic value near the MI-DW quantum phase transition. Here we fix \(t/\Omega= 0.05\) and vary \(g_{2}/\Omega\). Left panel: behavior at fixed system size \(L=300\), as a function of the distance r and for different values of \(g_{2}/\Omega= 1.3\), 1.35, 1.4, and 1.6. To minimize boundary effects, we chose the two points \((i, i+r)\) symmetrically with respect to the center of the array. Right panel: finite-size scaling close to the transition. Empty circles, filled circles, and triangles respectively are for \(g_{2} / \Omega= 1.4\) (DW), 1.35 (near the critical point), and 1.3 (MI phase). In the MI phase, \(C_{\mathrm{DW}}\) vanishes exponentially with L, according to the fits: \(C_{\mathrm{DW}}^{g_{2}=1.35} \approx0.241 \times e^{-0.032 L}\) and \(C_{\mathrm{DW}}^{g_{2}=1.3} \approx0.253 \times e^{-0.065 L}\). In the DW phase, \(C_{\mathrm{DW}}\) converges to a finite value. The inset shows such obtained asymptotic value \(\mathcal{O}_{\mathrm{DW}}\), as a function of \(g_{2}\).