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{\Large Homework 2}
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 Spectral functions and  Green's functions
in a perturbative expansion in $U$ to second order in $U$.

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1) Consider the symmetric ($\epsilon_f=-{U\over2}$) 
Anderson impurity model:
$$
H=\sum_{k\sigma}\epsilon_k c_{k\sigma}^{\dagger}c_{k\sigma}
+\sum_{k\sigma} (V_k c_{k\sigma}^{\dagger}f_{\sigma}+
V_k^*f_{\sigma}^{\dagger}c_{k\sigma})
+\epsilon_f(n_{\uparrow}+n_{\downarrow})+Un_{\uparrow}n_{\downarrow}
$$
Take the gybridization function 
$$
\Delta(i\omega_n)=
\sum_k{|V_k|^2\over i\omega_n-\epsilon_k}=-i \Gamma{\rm sign}{\omega_n}
$$
where $\Gamma=V^2\rho=1$ .



1)  Evaluate  numerically the Matsubara Green's function
$G(\tau)$ and plot it at  $\beta=1$, $1/10$ and $1/50$ at $U=2$.


2)  Evaluate numerically  
the $T=0$ spectral function and plot it  for $U=.5$, $1$, $1.5$, $2$, and $3$.

Carry out the calculation for parts 1 and  2 in    two schemes:

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a) bare perturbation $\Sigma=\Sigma(G^0,U)$  

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a) self-consistent perturbation theory $\Sigma=\Sigma_{skel}(G,U)$.


Comment on the differences between the results of the  two schemes.

3) Derive an expression for the spectral function at finite temperatures
and evaluate it for the same parameters as in part 2.



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