Low-temperature scaling laws in unconventional flat-band superconductors

In flat-band superconductors, the electron pairing is strongly enhanced so that the critical temperature scales linearly with the interaction strength. Identifying the governing pairing mechanism in flat-band superconducting systems is therefore a central task, which may be constrained by experimental probes via low-temperature scaling measurements. A key observable underlying the Meissner effect and the resulting divergent dc conductivity is the superfluid weight. While it is well established that the minimal quantum metric provides the dominant contribution to the superfluid weight in conventional superconductors with isolated flat bands, recent studies indicate that the unconventional pairing can generate additional nonlocal quantum geometric terms. This motivates us to derive the low-temperature scaling law of the superfluid weight in two-dimensional flat-band superconductors with sufficiently isolated bands. In particular, we consider the gap function with point or line nodes classified by the Weierstrass preparation theorem. Beyond the superfluid weight, we additionally deliver explicit low-temperature scaling laws of the order parameter, the tunneling conductance, the specific heat, the Sommerfeld coefficient, and the spin-lattice relaxation rate to provide complementary experimental discriminants of the underlying pairing symmetry. The implications of our results are also elucidated by applying them to a selection of superconducting states in $C_{6v}$-symmetric systems.