Color superconductors and holon metals from doping a Fractional Chern insulator

We develop a unified framework for metallic and superconducting phases obtained by doping a fractional Chern insulator (FCI) with $C=1/3$. Starting from the parton construction $c(\mathbf r)=f_1(\mathbf r)f_2(\mathbf r)f_3(\mathbf r)$, the low-energy theory has $SU(3)_{\mathrm{gauge}}\times SU(3)_{\mathrm{valley}}$ symmetry and nine Fermi pockets formed by charge-$-e/3$ holons $\psi_{ab}$, where $a$ and $b$ label color and valley. Viewing the holons as quarks connects this problem to color superconductivity in high-energy physics. Color-antisymmetric pairing produces a class of charge-$2e$ superconductors with angular momentum $L=3n$ and chiral central charge $c_-=m/2$, where $m$ is odd. Thus a gas of charge-$e/3$ anyons can enter a superconducting phase directly without binding. Particle--hole color--valley Higgs fields instead produce two $Z_3$ orthogonal metals with one or three pockets, transforming respectively as a singlet or triplet of $SU(3)_{\mathrm{valley}}$. A $U(1)^2$ holon metal with three identical pockets can preserve the triangular-lattice space group while reducing the emergent valley symmetry down to $S_3$. Its pairing instabilities include a gapped charge $2e$ $f-if$ superconductor and a gapless charge-$2e$ orthogonal superconductor with $\langle cc\rangle=0$ and a Bogoliubov Fermi surface at $\Gamma$. Finally, we discuss the possibility of a chemical-potential-tuned transition from the FCI to superconductivity and argue that all nine fermions may be required if the transition preserves the full emergent $SU(3)_v$ symmetry.