= Flat-band hybridization Fermi-velocity suppression
{title2=$v_F/v_F^{(1)}\simeq2^{1/3}|V|^2/[(2^{2/3}-1)^2E_d^2]$}
Mix a parabolic <energy band> $\epsilon_k=\hbar^2k^2/(2m)$ with a perfectly <flat band> at $E_d>0$ through momentum-independent $V$. At zero coupling suppose the two orbital sectors have equal populations, with the dispersive sector filled to $k_0=\sqrt{2mE_d}/\hbar$. At fixed total density and sufficiently low temperature, nonzero coupling fills only the monotonic lower hybridized branch, whose <Fermi wavevector> is $k_F=2^{1/3}k_0$, provided the available momentum-space states accommodate it. Its small slope at that radius gives the displayed ratio to $v_F^{(1)}=\hbar k_0/m$ for $|V|\ll E_d$. The result uses the singular perfectly flat limit and an empty upper band; it is not implied by <Fermi-volume conservation in two hybridized electron bands> for arbitrary pairs of dispersive bands. The zero-temperature limit must resolve the small hybridization energy scale.
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