  The plasmon excitations in proposed single- and double-component helical
liquid (HL) models are investigated within the random-phase approximation, by
calculating the density-density, spin-density and spin-spin waves. The effect
due to broken time-reversal symmetry on intraband-plasmon dispersion relation
in the single-component HL system is analyzed and compared to those of
well-known cases, such as conventional quasi-one-dimensional electron gases and
armchair graphene nanoribbons. The equivalence between the density-density wave
in the single-component HL to the coupled spin-density and density-density
waves in the double-component HL is shown here and explained, in addition to
the difference between intraband and interband-plasmon excitations in these two
systems. Since the two-component HL can physically be thought of as a Kramers
pair in two-dimensional topological insulators, our proposed single-component
HL model with broken time-reversal symmetry, which is an artificial construct,
can be viewed as an "effective" model in this sense and its prediction may be
verified in realistic systems in future experiments.
