  We use the tensor network algorithm to show evidences of a non-uniform chiral
spin-liquid (CSL) ground state in a frustrated spin-1/2 model on a square
lattice, in the regime of moat-like band structure of the lattice, i.e., a band
with infinitely degenerate energy minima attained along a closed contour in the
Brillouin zone. The analytical description of the state is given by the
effective field theory of a topological square-lattice fermionic Chern
insulator coupled to the Chern-Simons gauge field. The observed non-uniform CSL
has a substantial effect on a nearby free-fermion environment. We show that, in
the presence of arbitrarily small spin exchange interaction, the CSL can endow
a gauge-field-modulated effective interaction between the environmental
fermions. The induced effective interaction can be attractive within a
significant parameter region, leading to an instability towards d-wave
superconductivity in the fermionic bath.
