  Recent experiments reported an unusual nematic behavior of heavily hole-doped
pnictides $A$Fe$_{2}$As$_{2}$, with alkali $A$ = Rb, Cs. In contrast to the
$B_{2g}$ nematic order of the parent $Ae$Fe$_{2}$As$_{2}$ compounds (with
alkaline earth $Ae$ = Sr, Ba), characterized by unequal nearest-neighbor Fe-Fe
bonds, in the hole-doped systems nematic order is observed in the $B_{1g}$
channel, characterized by unequal next-nearest-neighbor Fe-Fe (diagonal
Fe-As-Fe) bonds. In this work, using density functional theory, we attribute
this behavior to the evolution of the magnetic ground state along the series
$Ae_{1-x}A_{x}$Fe$_{2}$As$_{2}$, from single stripes for small $x$ to double
stripes for large $x$. Our simulations using the reduced Stoner theory show
that fluctuations of Fe moments are essential for the stability of the
double-stripe configuration. We propose that the change in the nature of the
magnetic ground state is responsible for the change in the symmetry of the
vestigial nematic order that it supports.
