Majorana Zero Modes in the Lieb-Kitaev Model with Tunable Quantum Metric

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 Letter, we first construct 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 energy, the MZMs, which are supposed to be well localized at the ends of the 1D superconductor due to the flatness of the band, appear. 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 and induce ultra long range crossed Andreev reflections. This Letter 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.