  We theoretically study the effect of electron-electron interactions in
one-dimensional partially mixed helical states. These helical states can be
realized at the edges of two-dimensional topological insulators with partially
broken time-reversal symmetry, resulting in helical gapped states. Using the
bosonization method and renormalization group analysis, we identify weak gap,
crossover, and strong gap regimes in the phase diagram. We find that strong
electron-electron interaction mixes the helicity of the states, leading to the
relevant strong gap regime. We investigate the charge and spin density wave
correlation functions in different relevancy regimes of the gap mediated by
interactions, where in the case of strong repulsive interaction, the spin
density wave dominates the charge density wave. Additionally, employing the
Memory function technique, we calculate the effect of mixed helicity on the
charge transport in a sufficiently long edge. We find a non-uniform temperature
dependence for the charge conductivity in both the strong and weak gap regimes
with distinct features.
