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 $d+i{d}^{\prime }$-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 $d+i{d}^{\prime }$-wave superconducting state with a Chern number of 8, including the spin degree of freedom. The $\mathit{q}=\mathit{0}$ 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.

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• Dirac fermions
• Flat bands
• Geometric & topological phases
• Spin fluctuations
• Superconducting order parameter
• Superconductivity
• Quantum many-body systems
• Strongly correlated systems
• Topological superconductors
• Two-dimensional electron system
• Hubbard model