Electron-electron interactions in partially mixed helical states

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.