Correlation functions and characteristic lengthscales in flat band superconductors

The possibility of an unconventional form of high temperature superconductivity in flat band (FB) material does not cease to challenge our understanding of the physics in correlated systems. Recently, it has been argued that the coherence length in FB compounds could be decomposed into a conventional part of BCS type and a geometric contribution which characterises the FB eigenstates, the quantum metric. Here, we propose to address this issue in various FB systems and discuss whether the extracted characteristic lengthscales such as the size of the Cooper pairs obey this conjecture. It is found that the relevant lengthscales are less than one lattice spacing, weakly sensitive to the strength of the electron-electron interaction, and more importantly disconnected from the quantum metric.