Flat band Josephson junctions with quantum metric

In this work, we consider superconductor/flat band material/superconductor (S/FB/S) Josephs on 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>ξ</a:mi><a:mi>QM</a:mi></a:msub></a:math> of the flat bands. Microscopically, when <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>ξ</b:mi><b:mi>QM</b:mi></b:msub></b:math> 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, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>ξ</c:mi><c:mi>QM</c:mi></c:msub></c:math> 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 transistor which can carry a sizable and highly gate-tunable supercurrent.