  We investigate theoretically indirect exchange interaction between magnetic
impurities mediated by one-dimensional gapped helical states. Such states,
containing massive Dirac fermions, may be realized on the edge of a
two-dimensional topological insulator when time-reversal symmetry is weakly
broken. We find that the indirect exchange interaction consists of Heisenberg,
Dzyaloshinsky-Moriya, in-plane and out-of-plane Ising terms. These terms decay
exponentially when Fermi level lies inside the bandgap whereas the
Dzyaloshinsky-Moriya term has smallest amplitude. Outside the energy gap the
massive helical states modify oscillatory behaviors of the range functions so
that their periods decrease near the edge of band in terms of energy gap or
Fermi energy. In addition, the out-of-plane Ising term vanishes in the case of
zero-gap structure but its oscillation amplitude increases versus energy gap
and decreases as a function of Fermi energy whereas the oscillation amplitudes
of other components remain constant. Analytical results are also obtained for
subgap and over gap regimes. Furthermore, the effects of electron-electron
interactions are analyzed.
