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We show that the two-dimensional $\mathbb{Z}_2$ invariant for time-reversalinvariant insulators can be formulated in terms of the boundary-conditiondependence of the ground state wavefunction for both non-interacting andstrongly-correlated insulators. By introducing a family of quasi-singleparticle states associated to the many-body ground state of an insulator, weshow that the $\mathbb{Z}_2$ invariant can be expressed as the integral of acertain Berry connection over half the space of boundary conditions, providingan alternative expression to the formulations that appear in [Lee et al., Phys.Rev. Lett. $\textbf{100}$, 186807 (2008)]. We show the equivalence of thedifferent many-body formulations of the invariant, and show how they reduce toknown band-theoretic results for Slater determinant ground states. Finally, weapply our results to analytically calculate the invariant for the Kane-Melemodel with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. Thisrigorously establishes the topological nontriviality of the Kane-Mele modelwith HK interactions, and represents one of the few exact calculations of the$\mathbb{Z}_2$ invariant for a strongly-interacting system.

Computing the $\mathbb{Z}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems