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Consider a 1D solid with lattice spacing $a$. The inverse lattice vector is $K=1/a$. The Bloch expansion for the wave function can be written as

$$\psi(t,x)=\int d\omega dk\sum_{n\in\mathbb{Z}}\psi_n(\omega,k)\,e^{-i\omega t+i(k+nK)x},$$

where $k\in [-K/2,K/2]$, and $n$ is the label for the Brillouin zone. We can define a Green's function (think $\psi_n$ as an operator) as

$$G_{nn'}(\omega,k)=\langle\psi_n(\omega,k)\,\psi_{n'}^\dagger(\omega,k)\rangle.$$

  1. What is the meaning of $G_{nn'}$ for different $n$ and $n'$, and how are they related?
  2. What is the spectral function corresponding to the ARPES measurement, $\text{Im}(G_{00}+G_{11}+\cdots)$?
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2) For retarded Green's function, it is almost the same as ARPES measurement

(Only different with a normalization const, and some effects are difficult to including to Green function) $$ A_{apres} \simeq -\mathrm{Im}(\mathrm{Tr}\: G^R) = -\mathrm{Im} \left( G^R_{00}+G^R_{11} ...\right) $$

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