Towards a Ginzburg-Landau theory of the quantum geometric effect in superconductors*

Superconductivity in an isolated flat or quasiflat band system possesses novel features unseen in a conventional superconducting phase. In this paper, we establish a general framework of a Ginzburg-Landau (GL) theory to fully describe the quantum geometric effect in a multiband superconductor.  We derive a formula for the superfluid weight that depends on the quantum metric, BCS mean-field order parameter, and attractive interaction strength in the low-temperature limit.  We reveal a much enhanced upper critical field $H_{c2}$ inversely proportional to the quantum metric, which may hold even for a topologically trivial flatband. For a topologically nontrivial band, it can be bounded above by a quantity inversely proportional to the Chern number. We apply our theory to strained graphene with no net flux and a series of pseudo-Landau levels (pLL). An attractive

interaction makes the resultant superconducting robust on the zeroth pLLs because of the divergent density of states and no energy shift in a magnetic field. Furthermore, at the finite magnetic field, we propose an Fulde-Ferrell-Larkin-Ovchinnikov based on sign changes of quantum geometry induced superfluid weight. Lastly, appealing implications on twisted materials and some open questions are discussed.