Nodal pair density waves from a quarter-metal in crystalline graphene multilayers

Crystalline graphene heterostructures, namely, Bernal bilayer graphene (BBLG) and rhombohedral trilayer graphene (RTLG), for example, subject to perpendicular electric displacement fields, display a rich confluence of competing orders, resulting in a valley-degenerate, spin-polarized half-metal at moderate doping, and a spin- and valley-polarized (non-degenerate) quarter-metal at lower doping. Here we show that such a quarter-metal can be susceptible toward the nucleation of a unique spin- and valley-polarized superconducting ground state, accommodating \emph{odd-parity} (dominantly $p$ wave in BBLG and $f$ wave in RTLG) inter-layer Cooper pairs that break the translational symmetry, giving rise to a Kekul\'e (in BBLG) or columnar (in RTLG) pair density wave. Due to the trigonal warping in the normal state, the superconducting ground state produces three-fold rotationally symmetric isolated Fermi rings of normal fermions, which can manifest via linear in temperature scaling of the specific heat. We present scaling of the zero-temperature pairing amplitude and the transition temperature of such pair density wave in the presence of trigonally warped disconnected, annular, and simply connected Fermi rings in the normal state, subject to an effective attractive interaction within a mean-field approximation.