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Topological p+ip superconductivity in doped graphene-like single-sheet materials BC$_3$

We theoretically study exotic superconducting phases in graphenelike single-sheet material BC$_3$ doped to its type-II Van Hove singularity whose saddle point momenta are {\it not} time-reversal-invariant. From combined renormalization group analysis and RPA calculations, we show that the dominant superconducting instability induced by weak repulsive interactions is in the time-reversal-invariant $p+ip$ pairing channel because of the interplay among dominant ferromagnetic fluctuations, subleading spin fluctuations at finite momentum, and spin-orbit coupling. Such time-reversal-invariant $p+ip$ superconductivity has nontrivial $\mathbb{Z}_2$ topological invariant. Our results show that doped BC$_3$ provides a promising route to realize a genuine 2D helical $p+ip$ superconductor.