  Inspired by the newly discovered isomeric states in the rare-earth
neutron-rich nuclei, high-$K$ isomeric states in neutron-rich samarium and
gadolinium isotopes are investigated within the framework of the cranked shell
model (CSM) with pairing correlation treated by a particle-number-conserving
(PNC) method. The experimental multi-particle state energies and moments of
inertia are reproduced quite well by the PNC-CSM calculations. A remarkable
effect from the high-order deformation $\varepsilon_{6}$ is demonstrated. Based
on the occupation probabilities, the configurations are assigned to the
observed high-$K$ isomeric states. The lower $5^-$ isomeric state in $^{158}$Sm
is preferred as the two-proton state with configuration
$\pi\frac{5}{2}^{+}[413]\otimes\pi\frac{5}{2}^{-}[532]$. More low-lying
two-particle states are predicted. The systematics of the electronic quadrupole
transition probabilities, $B(E2)$ values along the neodymium, samarium,
gadolinium and dysprosium isotopes and $N=96,98,100,102$ isotones chains is
investigated to reveal the midshell collectivities.
