  Electrons with large kinetic energy have a superconducting instability for
infinitesimal attractive interactions. Quenching the kinetic energy and
creating a flat band renders an infinitesimal repulsive interaction the
relevant perturbation. Thus, flat band systems are an ideal platform to study
the competition of superconductivity and magnetism and their possible
coexistence. Recent advances in the field of twisted bilayer graphene highlight
this in the context of two-dimensional materials. Two dimensions, however, put
severe restrictions on the stability of the low-temperature phases due to
enhanced fluctuations. Only three-dimensional flat bands can solve the
conundrum of combining the exotic flat-band phases with stable order existing
at high temperatures. Here, we present a way to generate such flat bands
through strain engineering in topological nodal-line semimetals. We present
analytical and numerical evidence for this scenario and study the competition
of the arising superconducting and magnetic orders as a function of externally
controlled parameters. We show that the order parameter is rigid because the
quantum geometry of the Bloch wave functions leads to a large superfluid
stiffness. Using density-functional theory and numerical tight-binding
calculations we further apply our theory to strained rhombohedral graphite and
CaAgP materials.
