$\mathbb{Z}_2$ Topologically Obstructed Superconducting Order*

We propose a class of topological superconductivity in which the pairing order is $\mathbb{Z}_2$ topologically obstructed in a time-reversal invariant system in three dimensions. When two Fermi surfaces are related by time-reversal and mirror symmetries, such as those in a $\mathbb{Z}_2$ Dirac semimetal, the inter-Fermi-surface pairing in the weak-coupling regime inherits the band topological obstruction. As a result, the pairing order cannot be well-defined over the entire Fermi surface and forms a time-reversal invariant generalization of U($1$) monopole harmonic pairing. A tight-binding model of the $\mathbb{Z}_2$ topologically obstructed superconductor is constructed based on a doped $\mathbb{Z}_2$ Dirac semimetal and exhibits nodal gap function. At an open boundary, the system exhibits a time-reversal pair of topologically protected surface states.