Effect of superconductivity on the shape of flat bands

For the first time, basing both on experimental facts and our theoretical consideration, we show that Fermi systems with flat bands should be tuned with the superconducting state. 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 $V_F\propto T_c\propto 1/N_s(0)$,where$N_s(0)$ is the density of states at the Fermi level. We show that the high-$T_c$ compounds $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{r}}_2\mathrm{C}\mathrm{a}\mathrm{C}{\mathrm{u}}_2{\mathrm{O}}_{8+\mathrm{x}}$ exhibit the same behavior. Such observation is a challenge to theories of high-$T_c$ superconductivity, since $V_F$ is negatively correlated with $T_c$, for $T_c\propto 1\mathrm{/}{V}_F\propto N_s(0)$. We show that the theoretical idea of forming flat bands in strongly correlated Fermi systems can explain this behavior and other experimental data collected on both $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{r}}_2\mathrm{C}\mathrm{a}\mathrm{C}{\mathrm{u}}_2{\mathrm{O}}_{8+\mathrm{x}}$ and twisted bilayer graphene. Our findings place stringent constraints on theories describing the nature of high-$T_c$ superconductivity and the deformation of flat band by the superconducting phase transition.