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.
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The PDF file of the response to the referees has been merged with the new PDF version of the manuscript.
List of changes
* Appendix C has been replaced * Section "Connection with recent studies" have been entirely rewritten * A new figure has been added in the main text (Fig. 7) * A new figure has been added in the new version of the appendix C (Fig. 8) * Changes in the text appear in blue for this second round. We recall that the changes in red correspond to the first round. * We have added three new references