Flat Band Josephson Junctions with Quantum Metric

In this work, we consider superconductor/flat band material/superconductor (S/FB/S) Josephson junctions (JJs) where the flat band material possesses isolated flat bands with exactly zero Fermi velocity. Contrary to conventional S/N/S JJs where the critical Josephson current vanishes when the Fermi velocity goes to zero, we show in this work that the critical current in the S/FB/S junction is controlled by the quantum metric length $\xi_\mathrm{QM}$ of the flat bands. Microscopically, when $\xi_\mathrm{QM}$ of the flat band is long enough, the interface bound states originally localized at the two S/FB, FB/S interfaces can penetrate deeply into the flat band material and hybridize to form Andreev bound states (ABSs). These ABSs are able to carry long range and sizable supercurrents. Importantly, $\xi_\mathrm{QM}$ also controls how far the proximity effect can penetrate into the flat band material. This stands in sharp contrast to the de Gennes' theory for S/N junctions which predicts that the proximity effect is expected to be zero when the Fermi velocity of the normal metal is zero. We further suggest that the S/FB/S junctions would give rise to a new type of resonant Josephson transistors which can carry sizable and highly gate-tunable supercurrent.