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Electrons with large kinetic energy have a superconducting instability forinfinitesimal attractive interactions. Quenching the kinetic energy andcreating a flat band renders an infinitesimal repulsive interaction therelevant perturbation. Thus, flat band systems are an ideal platform to studythe competition of superconductivity and magnetism and their possiblecoexistence. Recent advances in the field of twisted bilayer graphene highlightthis in the context of two-dimensional materials. Two dimensions, however, putsevere restrictions on the stability of the low-temperature phases due toenhanced fluctuations. Only three-dimensional flat bands can solve theconundrum of combining the exotic flat-band phases with stable order existingat high temperatures. Here, we present a way to generate such flat bandsthrough strain engineering in topological nodal-line semimetals. We presentanalytical and numerical evidence for this scenario and study the competitionof the arising superconducting and magnetic orders as a function of externallycontrolled parameters. We show that the order parameter is rigid because thequantum geometry of the Bloch wave functions leads to a large superfluidstiffness. Using density-functional theory and numerical tight-bindingcalculations we further apply our theory to strained rhombohedral graphite andCaAgP materials.

Designing three-dimensional flat bands in nodal-line semimetals