Geometric Contribution to the Electron-Phonon Coupling Constant in the Checkerboard Model and Hyperdiamond Model

Recent theoretical studies on electron–phonon coupling (EPC)—a key ingredient in the emergence of diverse quantum phases such as superconductivity and charge-density waves—have revealed a distinct contribution to its strength arising from the quantum geometry of electronic bands. Here we investigate the quantum geometric contribution, characterized by the quantum metric, to the dimensionless EPC constant in the checkerboard and hyperdiamond models, under the Gaussian approximation. We show that in the checkerboard model, the geometric contribution can be tuned independently while keeping the energetic part fixed, demonstrating the possibility of enhancing EPC strength through quantum geometry. In the hyperdiamond model, the geometric contribution accounts for roughly 50% of the total EPC constant across all spatial dimensions, generalizing the d=2 result for graphene to higher dimensions.