  We find that a $Z$-string is stable in a wide range of parameter space of the
two Higgs doublet model due to a split into a pair of two topological
$Z$-strings with fractional $Z$ fluxes. This configuration, a bound state of
the two strings connected by a domain wall, is called a vortex molecule.
Although the vortex molecule has no net topological charge, the locally induced
topological charge density is polarized, namely distributed positively around
one constituent string and negatively around the other constituent string,
leading to the stability of the molecule. We numerically show that the vortex
molecule is indeed a stable solution of the equation of motions in a much wider
parameter space of the model than the usual axially symmetric $Z$-string in the
Standard Model and the two Higgs doublet model, although it is not the case for
experimental values of the parameters.
