Possible 'symmetry-imposed' near-nodal two-dimensional p-wave pairing in Sr$_2$RuO$_4$

One key feature of the multi-orbital superconducting Sr$_2$RuO$_4$ is the presence of nodal or near-nodal quasiparticle excitations revealed in a wide variety of experiments. Typically, a nodal gap structure in a two-dimensional model would be inconsistent with the chiral or helical p-wave interpretations. However, we demonstrate that true gap nodes may emerge along certain high-symmetry directions on the quasi-one-dimensional Fermi surfaces, if the multi-orbital chiral or helical p-wave pairings acquire peculiar forms wherein the $d_{xz}$ and $d_{yz}$ orbitals develop $k_y$- and $k_x$-like pairings, respectively. Spin-orbit coupling $\eta$ induces a near-nodal gap of order $\eta^2/W^2 \Delta_0$, where $\Delta_0$ is the gap amplitude and $W$ roughly the bandwidth. Provided the aforementioned pairing is predominant, the near-nodal gap structure is robust upon the inclusion of other multi-orbital pairings that share the same symmetries. In light of the recent experimental progresses, our proposal suggests that two-dimensional p-wave pairings may still be viable candidate ground states for Sr$_2$RuO$_4$. A near-nodal helical p-wave order, for example, would also be consistent with the substantial drop in the NMR Knight shift under an in-plane magnetic field.