@sk06; @sm09-2; @sp07] In this mechanism hole-pocket Fermi surfaces around the nodal directions coexist with a zero surface around $(\pi,\pi)$. Because Im$\S$ is large around the zero surface, the pockets lose the spectral intensity more on the side closer to the zero surface, leaving arc-like spectra on the opposite side. This is reproduced in Figs. \[fig:pocket\](a) and (b). Figure \[fig:pocket\](a) shows the underlying pole-zero structure calculated by the CDMFT+ED at $T=0$ without any substantial smearing to the singularities, where a clear hole-pocket Fermi surface coexists with a zero surface. When we introduce a smearing to the singularities (see Sec. \[ssec:ed\]), we obtain the spectral map, Fig. \[fig:pocket\](b), in which the arc-like structure is observed.

We confirm the zero mechanism by directly calculating a spectral weight at finite temperatures by employing the CTQMC method as the solver for the CDMFT. To obtain the spectrum without through any analytic continuation procedure which inevitably suffers from a large error bar, we calculate $$\begin{aligned}
\label{eq:ghb}
-\b G(\Vec{k},\t=\beta/2)=\frac{\b}{2}
\int_{-\infty}^{\infty}\frac{A(\Vec{k},\w)}{\cosh(\b\w/2)}d\w\end{aligned}$$ as a function of $\Vec{k}$. This quantity is an integral of the spectral weight over a width $\sim T$ around $\w=0$ and approaches $A(\Vec{k},\w=0)$ for large $\beta$, so that gives an estimate of $A(\Vec{k},\w=0)$ at low temperatures.[@footnote2] In Figs. \[fig:pocket\](c) and (d) we plot the results at $T=0.025t$. For $n=0.95$ \[Fig. \[fig:pocket\](c)\] we see an arc at a location similar to the one in Fig. \[fig:pocket\](b) while for $n=0.9$ \[Fig. \[fig:pocket\](d)\] the spectra extend to the antinodal regions. The doping evolution of the spectra is qualitatively consistent with that obtained with the CDMFT+ED in Fig. 2(c)-(e) in Ref. . The qualitative agreement between the CDMFT+ED result with broadening $\eta$ and the CDMFT+CTQMC result at finite temperatures supports that the phenomenological smearing 