  Recent experimental study unveiled highly unconventional phenomena in the
superconducting twisted bilayer graphene (TBG) with ultra flat bands, which
cannot be described by the conventional BCS theory. For example, given the
small Fermi velocity of the flat bands, the superconducting coherence length
predicted by BCS theory is more than 20 times shorter than the measured values.
A new theory is needed to understand many of the unconventional properties of
flat band superconductors. In this work, we establish a Ginzburg-Landau (GL)
theory from a microscopic flat band Hamiltonian. The GL theory shows how the
properties of the physical quantities such as the critical temperature, the
superconducting coherence length, the upper critical field and the superfluid
density are governed by the quantum metric of the Bloch states. One key
conclusion is that the superconducting coherence length is not determined by
the Fermi velocity but by the size of the optimally localized Wannier functions
which is limited by quantum metric. Applying the theory to TBG, we calculated
the superconducting coherence length and the upper critical fields. The results
match the experimental ones well without fine tuning of parameters. The
established GL theory provides a new and general theoretical framework for
understanding flat band superconductors with quantum metric.
