Impact of flowing three-body operators on electric quadrupole observables in the in-medium similarity renormalization group

We investigate the impact of flowing three-body operators on electric quadrupole ($E2$) observables within the in-medium similarity renormalization group (IMSRG) framework. While the standard VS-IMSRG(2) truncation provides reliable ab initio predictions for a wide range of nuclear properties, it significantly underpredicts collective $E2$ observables, such as transition strengths and quadrupole moments. In this work, we extend the IMSRG framework to include consistent three-body contributions in both the Hamiltonian and the $E2$ operator transformation, denoted as IMSRG(3). Due to computational constraints, we apply a truncation (IMSRG(3n7)) that retains only commutators scaling as $N^7$ or less. We present results for $^{17}$O, $^{12}$C, and $^{28}$Si, demonstrating that IMSRG(3n7) provides modest but systematic improvements in $E2$ observables compared to IMSRG(2). These results suggest that the approximations in IMSRG(3) can slightly improve predictions of collective observables compared to IMSRG(2), but they still fail to reproduce the strong experimental $E2$ transitions.