to (ii), our results do not show strong intensity around $(\frac{\pi}{2},\frac{\pi}{2})$; this will be attributed to the absence of the third-neighbor transfer integral $t''$ in the present calculation, as discussed below.

Second, Figs. \[fig:edope\](g)-(i) show the CDMFT+ED results for $t'=-0.4t$ and $U=12t$ at $n=1.14$, $1.17$, and $1.19$, respectively. We see that a zero surface exists at the Fermi level, surrounding $(0,0)$. For low dopings electron-pocket Fermi surfaces reside around the antinodal points. As the doping is increased, the pockets expand. Then they merge around the nodal point and change the topology into two Fermi surfaces, one around $(\pi,\pi)$ and the other around $(0,0)$. Further doping makes the inner Fermi surface annihilate in pair with the zero surface, resulting in the Fermi liquid with the single large Fermi surface around $(\pi,\pi)$.[@footnote3] Thus in electron doping there occur, at least, two phase transitions, the Lifshitz transition [@ko07] and a pole-zero annihilation transition, on the way from the Mott insulator to the Fermi liquid.

The presence of the zero surface accounts for the weak spectral intensity around the nodal direction in Figs. \[fig:edope\](d) and (e). Then how is the intensity around the node in Fig. \[fig:edope\](b) and (c) explained? To answer this we consider how the electronic structure, Fig. \[fig:rkw\_tp\](d), changes with $t''$, which is not included in our calculations based on the $2\times2$ cluster. Although $|t''|$ is usually several times smaller than $|t'|$ and therefore negligible for discussing global electronic structures, it can be crucial to understand the nodal intensity. This is because around $(\frac{\pi}{2},\frac{\pi}{2})$ the pole surface at $\w\simeq -0.15t$ in Fig. \[fig:rkw\_tp\](d) moves up with $-t'/t$, as seen in comparison with Fig. \[fig:rkw\_tp\](c), and is located just below $E_{\text{F}}$ for $-t'/t=0.4$ (not shown). Since to the first approximation $t''(>0)$ elevates dispersions around $(\frac{\pi}{2},\frac{\pi}{2})$