Systematics of E2 strength in the sd-shell with the valence-space in-medium similarity renormalization group

Background: Recent developments in {\it ab initio} nuclear theory demonstratepromising results in medium- to heavy-mass nuclei. A particular challenge formany of the many-body methodologies, however, is an accurate treatment of theelectric-quadrupole, $E2$, strength associated with collectivity. Purpose: Thevalence-space in-medium similarity renormalization group (VS-IMSRG) is aparticularly powerful method for accessing medium- and high-mass nuclei but hasbeen found to underpredict $E2$ strengths. The purpose of this work is toevaluate the isospin dependence of this underprediction. Methods: We perform asystematic comparison of valence-space in-medium similarity renormalizationgroup (VS-IMSRG) calculations with available literature. We make use ofisoscalar and isovector contributions to the $E2$ matrix elements to assessisoscalar and isovector contributions to the missing strength. Results: It isfound that the $E2$ strength is consistent throughout$T_z=\left|\frac{1}{2}\right|$, $T_z=\left|1\right|$,$T_z=\left|\frac{3}{2}\right|$ and $T_z=2$ pairs within the $sd$-shell.Furthermore, no isovector contribution to the deficiency is identified.Conclusions: A comparison with toy-models and coupled-cluster calculations isused to discuss potential origins of the missing strength, which arises frommissing many-particle, many-hole excitations out of the model space. Theabsence of any significant isovector contribution to the missing $E2$ strengthindicates that the $E2$ strength discrepancy, and therefore any correction, islargely independent of the isospin of the nuclei in question