Magnon-Mediated Superconductivity in a 2D Itinerant Ferromagnet with Weak Easy-plane Magnetic Anisotropy

Motivated by recent observations of superconductivity in a quarter-metal state of spin- and valley- polarized graphene multilayers, we investigate pairing within a ferromagnetic phase of a single-valley model of itinerant two-dimensional (2D) electrons with Hubbard-type interaction and no artificial high-energy cutoff. In 2D, the Stoner transition is first-order into a fully-polarized state wherein the only gapless collective excitations are transverse magnons. We find that in a spin-SU(2) symmetric model, this magnon-mediated pairing interaction between equal-spin fermions vanishes at $T=0$. We show that a small easy-plane magnetic anisotropy $\Omega_0 \ll E_F$, where $E_F$ is the Fermi energy, breaks the SU(2) symmetry and generates an attractive interaction for equal-spin $p-$wave pairing. We explicitly derive the corresponding coupling constant $\lambda_p$ as the scaling function of both the relative strength of the easy-plane anisotropy, $\Omega_0/E_F$, and the proximity to the ferromagnetic transition. While $\lambda_p$ is parametrically small in $\Omega_0/E_F$ deep inside the ferromagnetic phase, it becomes enhanced near the ferromagnetic transition, reaching order unity regardless of how small $\Omega_0/E_F$ is. This mechanism yields a sizable $T_c$, peaked near the onset of ferromagnetism.