Electron removal spectral function ($f(ϵ)A(\mathbf{k},ϵ)$)

The Electron single-particle spectral function $A(\mathbf{k},ϵ)=-\mathrm{Im}{G}^R(\mathbf{k},ϵ)/\pi$, in a convention normalized to unit energy integral per orbital and spin, describes addition and removal energies. For the fermionic retarded Green function, $G^R(\mathbf{k},t)=-i\Theta (t)⟨\{c_{\mathbf{k}}(t),c_{\mathbf{k}}^{†}(0)\}⟩$, energy is referenced to the chemical potential. The occupied removal spectrum is $f(ϵ)A(\mathbf{k},ϵ)$, with Fermi-Dirac distribution $f$. In the noninteracting limit, $A$ is a delta peak at the band energy; interactions can shift, broaden, and redistribute its weight.