  We explore the Dirac fermionic and complex scalar dark matter in the
framework of a hidden $\mathrm{U}(1)_\mathrm{X}$ gauge theory with kinetic
mixing between the $\mathrm{U}(1)_\mathrm{X}$ and $\mathrm{U}(1)_\mathrm{Y}$
gauge fields. The $\mathrm{U}(1)_\mathrm{X}$ gauge symmetry is spontaneously
broken due to a hidden Higgs field. The kinetic mixing provides a portal
between dark matter and standard model particles. Besides, an additional Higgs
portal can be realized in the complex scalar case. Dark matter interactions
with nucleons are typically isospin violating, and direct detection constraints
can be relieved. Although the kinetic mixing has been stringently constrained
by electroweak oblique parameters, we find that there are several available
parameter regions predicting an observed relic abundance through the thermal
production mechanism. Moreover, these regions have not been totally explored in
current direct and indirect detection experiments. Future direct detection
experiments and searches for invisible Higgs decays at a Higgs factory could
further investigate these regions.
