  We propose to implement the softly broken Z_2 symmetry in the usual two Higgs
doublet model (2HDM) to spontaneously broken local U(1)_H gauge symmetry, and
show that the resulting phenomenology can be very rich and is distinctly
different from the usual 2HDMs. Likewise, the exact Z_2 symmetry in ordinary
inert doublet model (IDM) could be a remnant of spontaneously broken local
U(1)_H symmetry stabilizing dark matter (DM). In this case, new channels for DM
pair annihilation into U(1)_H gauge boson(s) open up, allowing the DM mass
below \lesssim 40 GeV, unlike the usual IDM.
