  Over the past decade the in-medium similarity renormalization group (IMSRG)
approach has proven to be a powerful and versatile ab initio many-body method
for studying medium-mass nuclei. So far, the IMSRG was limited to the
approximation in which only up to two-body operators are incorporated in the
renormalization group flow, referred to as the IMSRG(2). In this work, we
extend the IMSRG(2) approach to fully include three-body operators yielding the
IMSRG(3) approximation. We use a perturbative scaling analysis to estimate the
importance of individual terms in this approximation and introduce truncations
that aim to approximate the IMSRG(3) at a lower computational cost. The
IMSRG(3) is systematically benchmarked for different nuclear Hamiltonians for
${}^{4}\text{He}$ and ${}^{16}\text{O}$ in small model spaces. The IMSRG(3)
systematically improves over the IMSRG(2) relative to exact results.
Approximate IMSRG(3) truncations constructed based on computational cost are
able to reproduce much of the systematic improvement offered by the full
IMSRG(3). We also find that the approximate IMSRG(3) truncations behave
consistently with expectations from our perturbative analysis, indicating that
this strategy may also be used to systematically approximate the IMSRG(3).
