um_{ij\in \text{C}}\sum_n 
     \left|\left[\hat{g}_0(i\w_n)\right]_{ij}
     -\left[\hat{g}_{0,N_{\text{B}}}(i\w_n)\right]_{ij}\right|^2 
     e^{-\w_n/t},
\label{eq:distance}\end{aligned}$$ where $\hat{g}_{0,N_{\text{B}}}$ is the non-interacting Green’s function for the effective Hamiltonian and we have introduced the exponential weight factor $e^{-\w_n/t}$ with $\w_n\equiv (2n+1)\pi/\b'$ to reproduce more precisely the important low-frequency part. We have examined several other types of distance functions and confirmed that qualitative feature of the results obtained in this paper does not depend on the choice.

An advantage of the Lanczos method is that Green’s function is obtained as a function of real frequency $\w$ by the continued-fraction expansion, $$\begin{aligned}
G(\w)&=\frac{\langle 0| c c^\dagger | 0\rangle}
   {\w+i\eta-a_0^>-\displaystyle{ \frac{{b_1^>}^2}{\w+i\eta-a_1^>
    - \displaystyle{\frac{{b_2^>}^2}{\w+i\eta-a_2^>-\cdots}}}}}\nonumber\\
  &+\frac{\langle 0| c^\dagger c | 0\rangle}
   {\w+i\eta-a_0^<-\displaystyle{\frac{{b_1^<}^2}{\w+i\eta-a_1^<
    -\displaystyle{\frac{{b_2^<}^2}{\w+i\eta-a_2^<-\cdots}}}}},
\label{eq:cf}\end{aligned}$$ where $|0\rangle$ is the ground-state vector and the coefficients $a_i^{>,<}$ and $b_i^{>,<}$ are the elements of the tridiagonal matrix appearing in the Lanczos algorithm.[@gb87] We take account of up to 2000th order in the expansion. A small positive $\eta$ is introduced to satisfy the causality. In principle, $\eta$ is taken to be infinitesimal, but in practice, it is useful to consider $\eta$ as a parameter, which serves as a resolution in energy or mimics an infinite-size effect not incorporated into the ED calculation.

In the following study, we pursue a further benefit of the parameter $\eta$: We use $\eta$ as a mimic of the source of incoherence in the electronic structure, such as thermal or impurity scattering. This is based on the observation that $\eta$ does not substantially change the location of poles and zeros of $G$ but changes only t