Spin-fluctuation-mediated chiral $d+i{d}^{\mathrm{\prime }}$-wave superconductivity in the $\alpha$-${\mathcal{T}}_3$ lattice with an incipient flat band

We study anisotropic superconductivity in the nearly quarter-filled $\alpha$-${\mathcal{T}}_3$ lattice. We analyze an extended Hubbard model with off-site attractive interactions within the mean-field framework and find two distinct chiral $d+i{d}^{\mathrm{\prime }}$-wave superconducting phases characterized by different Chern numbers. We further investigate the superconducting mechanism mediated by spin fluctuations arising from purely repulsive interactions by applying the fluctuation-exchange (FLEX) approximation to the Hubbard model. The gap symmetry obtained by solving the linearized Eliashberg equation is $d$-wave, which corresponds to a $d+i{d}^{\mathrm{\prime }}$-wave superconducting state with a Chern number of $8$, including the spin degree of freedom. The $\mathbf{q}=\mathbf{0}$ antiferromagnetic spin fluctuation, which possesses the largest spectral weight at finite energies arising from the incipient flat band, gives rise to an effective spin-singlet pairing glue between rim sites.