  We study superconductivity in the hadron-quark mixed phase of planar quantum
chromodynamics (QCD) within the large $N$ limit of a Gross-Neveu model modified
by a repulsive vector term. At high densities, the combination of scalar
attraction and repulsive space-like part of the vector interaction squeezes
quarks into baryonic composite states, i.e., Dirac fermions with even numbers
of bosonic vortices attached. The time-like vector component induces Cooper
pairing between these Fermi surface modes. Remarkably, at zero temperature,
competition between the quark density and mass destroys superconductivity via a
Berezinskii-Kosterlitz-Thouless (BKT) phase transition driven by diverging
chiral quantum fluctuations near criticality. Dissolution of logarithmically
bound singlet diquarks is catalyzed by in-plane chiral mixing associated with
$\mathbb{Z}_2 \otimes \mathbb{Z}_2 \to \mathbb{Z}_2$ chiral symmetry breaking
of the Fermi surface into a transverse spin-polarized triplet ground state. We
calculate the QCD phase diagram for quark chemical potential above the baryon
mass based purely on Fermi surface considerations and find good agreement with
results obtained by other methods. We address similarities between our quantum
BKT transition and those found using holographic techniques.
