  The relation between band topology and Majorana zero energy modes (MZMs) in
topological superconductors had been well studied in the past decades. However,
the relation between the quantum metric and MZMs has yet to be understood. In
this work, we first introduce a three band Lieb-like lattice model with an
isolated flat band and tunable quantum metric. By introducing nearest neighbor
equal spin pairing, we obtain the Lieb-Kitaev model which supports MZMs. When
the Fermi energy is set within the flat band, the MZMs are supposed to be
well-localized at the ends of the 1D superconductor due to the flatness of the
band. On the contrary, we show both numerically and analytically that the
localization length of the MZMs is controlled by a length scale defined by the
quantum metric of the flat band, which we call the quantum metric length (QML).
The QML can be several orders of magnitude longer than the conventional BCS
superconducting coherence length. When the QML is comparable to the length of
the superconductor, the two MZMs from the two ends of the superconductor can
hybridize. When two metallic leads are coupled to the two MZMs, crossed Andreev
reflection probability can nearly reach the maximal theoretical value. This
work unveils how the quantum metric can greatly influence the properties of
MZMs through the QML and the results can be generalized to other topological
bound states.
