= Quasiparticle residue
{title2=$Z$}
The <quasiparticle residue> is the coherent pole weight of the single-particle <Green's function>. Its coherent spectral term $Z\delta(E-\xi_p)$ contributes $Z\Theta(-\xi_p)$ to the zero-<temperature> <momentum> occupation. A smooth remainder leaves a jump $Z$ at the <Fermi surface>; <Landau quasiparticle> occupations have a unit step.<On the Theory of the Fermi Liquid by Lev Davidovich Landau>[p.70, equations (4)–(6); integrate the pole spectral weight]{ref}
For the local illustrative distribution in the figure, put $x=(p-p_F)/p_F$ and
$$
n_{\rm particle}(x)=0.65\Theta(-x)+0.35\left[\frac12-0.2\tanh(2x)\right].
$$
$\Theta$ is the <Heaviside step function>. Its one-sided limits are $0.825$ and $0.175$, giving $Z=0.65$. The smooth term is stipulated; this local curve does not specify a normalized distribution over all momenta. In particular, $Z$ is neither the fraction of atoms above the <Fermi surface> nor $m/m^*$.
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