  In the present work, we have reported shell model results for open shell
nuclei Ne, Mg and Si isotopes with $10 \leq N \leq 20$ in $sd$-shell model
space. We have performed calculations in $sd$ shell with two $ab~initio$
approaches: in-medium similarity renormalization group (IM-SRG) and
coupled-cluster (CC) theory. We have also performed calculations with
phenomenological USDB interaction and chiral effective field theory based CEFT
interaction. The results for rotational spectra and $B(E2;2_1^+\rightarrow
0_1^+)$ transitions are reported for even-mass isotopes. The IM-SRG and CC
results are in reasonable agreement with the experimental data except at $N$
=20. This demonstrates a validity of $ab~initio$ description of deformation for
doubly open-shell nuclei for $sd$ shell. To see the importance of $pf$
orbitals, we have also compared our results with SDPF-MU interaction by taking
account of $2p-2h$ and $4p-4h$ configurations in $sd$-$pf$-shell model space.
