Effect of superconductivity on the shape of flat band

Recent experimental measurements on magic-angle twisted bilayer graphene of the Fermi velocity $V_F$ as a function of the temperature $T_c$ of superconduction phase transition have revealed that $V_F$ is proportional to $T_c$, $T_c\propto V_F\propto 1/N_s(0)$, where $N_s(0)$ is the density of states at the Fermi level. Such a behavior is a challenge to the theories of high-$T_c$ superconductivity, since $V_F$ is always negatively correlated with $T_c$, for $T_c\propto 1/V_F\propto N_s(0)$. We show that the theoretical idea of forming flat bands in a Fermi system, that has been proposed three decades ago and revisited recently, can explain this behavior and the other experimental facts collected on twisted bilayer graphene with flat bands. Our findings place strong constraints on theories describing the nature of high-$T_c$ superconductivity and the flexible behavior of flat bands.