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Topological properties of possible Weyl superconducting states of URu$_\mathbf{2}$Si$_\mathbf{2}$

We show that the current thermodynamic measurements in the superconducting phase of $\mathrm{U}\mathrm{Ru}_2\mathrm{Si}_2$ are compatible with two distinct singlet chiral paired states $k_z(k_x \pm i k_y)$ and $(k_x \pm i k_y)^2$. Despite possessing similar low temperature thermodynamic properties, these two pairings are topologically distinguished by their respective orbital angular momentum projections along the c-axis, $m=\pm 1$ and $m=\pm 2$. The point nodes of these states act as the monopoles and the anti-monopoles of the Berry's gauge flux of charge $\pm m$, which are separated in the momentum space along the $c$ axis. As a result, the Berry's flux through the $ab$ plane equals $m$. Consequently, the point nodes of $k_z(k_x+i k_y)$ and $(k_x \pm ik_y)^2$ states respectively realize the Weyl and the double-Weyl fermions, with chemical potential exactly tuned at the Fermi point, due to the charge conjugation symmetry. These topologically nontrivial point nodes, give rise to $m$ copies of protected spin degenerate, chirally dispersing surface states on the $ca$ and the $cb$ planes, which carry surface current, and their energies vanish at the Fermi arcs. In contrast, a line node acts as the momentum space vortex loop, and gives rise to the zero energy, dispersionless Andreev bound states on the surfaces parallel to the plane enclosed by the line node. The Berry's flux through the $ab$ plane gives rise to anomalous spin Hall and thermal Hall conductivities, and various magnetoelectric effects. A clear determination of the bulk invariant can only be achieved by probing the pairing symmetry via a corner Josephson junction measurement, and Fourier transformed STM measurements of the Fermi arcs. Therefore, we identify $\mathrm{U}\mathrm{Ru}_2\mathrm{Si}_2$ as a promising material for realizing gapless topological superconductivity in three spatial dimensions.