In a conventional metal, Cooper instability is unconditional: any attractive interaction, no matter how weak, produces a superconducting ground state below some critical temperature. The Fermi surface provides a perfect nesting condition — the density of states at the Fermi level is finite, and the logarithmic divergence in the particle-particle channel means even infinitesimal attraction wins. This is the BCS result, and it's why superconductivity is surprisingly common.
In semimetals with vanishing density of states at the Fermi level — Dirac and Weyl systems where the bands touch at discrete points — the situation changes. The vanishing DOS means the particle-particle susceptibility no longer diverges logarithmically. Cooper pairing requires a threshold interaction strength: the attraction must exceed a critical value before superconductivity appears. Below the threshold, the system remains a semimetal regardless of temperature.
Zang and Wang (arXiv 2602.22772, February 2026) analyze Cooper instability in fractional Dirac semimetals — systems where the dispersion near the band touching point goes as |k|^α for a fractional exponent α, generalizing the linear (α=1) Dirac case. The fractional exponent controls the density of states: larger α means higher DOS at the touching point, which means easier Cooper pairing. The threshold interaction strength decreases with increasing α, interpolating continuously between the semimetal regime (threshold exists) and the metal regime (no threshold).
The transfer momentum parameter adds a second axis. Momentum space splits into two regions: one where Cooper instability is permitted (the interaction exceeds the local threshold) and one where it is suppressed. The phase diagram is a map of where superconductivity can and cannot nucleate in momentum space.
Disorder introduces the unexpected result. Certain types of disorder — those that scatter between states of matching symmetry — lower the threshold for Cooper pairing. They promote superconductivity, expanding the region of momentum space where pairing is allowed. Other disorder types raise the threshold, suppressing pairing. The two effects compete when both disorder types coexist, and the suppressive types generally win.
The promotion mechanism is specific: the “right” disorder enhances the effective density of states at the Fermi level by broadening the band touching point. A sharp Dirac cone has vanishing DOS; a disordered Dirac cone has finite DOS from the disorder-induced broadening. The broadening provides the finite density of states that the BCS mechanism requires. The disorder fills in the gap that was preventing superconductivity.
The clean system resists pairing because its electronic structure is too sharp. The dirty system superconducts because the disorder blunts the sharpness. The threshold was in the precision, not in the attraction.