case of $N_s = 100$ and $N_e = 84$ is of the order of magnitude of the BCS superconducting condensation energy $\sim$ (state density)$\times \Delta^2 \approx 
0.0010$ since the state density per site per spin around this density is approximately equal to $1/2\pi$ in units of $t$.

In order to check the superconducting nature of the ground state with a finite value of $\Delta$, the correlation functions of BCS pair operators were calculated. Superconducting pair correlation functions $D_{\alpha \beta}(l), \alpha, \beta = x, y$, are defined as: $$D_{\alpha \beta}(l) = < \Delta_{\alpha}^{\dag}(i+l,j)
           \Delta_{\beta}(i,j) >  , \label{eq:Dab}$$ where $\Delta_{\alpha}(i,j), \alpha = x, y$, denote the annihilation operators of singlet electron pairs staying on n.n. sites as: $$\begin{aligned}
\Delta_x(i,j) & = & c_{ij\downarrow}c_{i+1,j\uparrow}
                   -  c_{ij\uparrow}c_{i+1,j\downarrow} ,  \\
\Delta_y(i,j) & = & c_{ij\downarrow}c_{i,j+1\uparrow}
                   -  c_{ij\uparrow}c_{i,j+1\downarrow} ,\end{aligned}$$ where $c_{ij\sigma}$ means the annihilation operator at site $(i, j)$ with spin $\sigma$. The average $<\ldots>$ is defined in eq. (\[eq:average\]). The result for a state close to the minimum $E_g$ point in Fig. \[fig:DEg-D\], i.e., $\Delta = 0.078$, $\mu = -0.428$ and $g = 0.30$, is shown in Fig. \[fig:cor1\]. The vertical scale is enlarged to highlight the long range parts. The correlation extends over the 