Interaction effects are particularly important in the 1D helical edge states of the 2D topological insulators, and in the Majorana edge modes of topological superconductors. Although the single particle backscattering is forbidden by time-reversal symmetry determined by the Z2 topology, the two-particle correlated backscattering processes are allowed by the time-reversal symmetry, and can destabilize the edge states by opening the gap with spontaneously developing magnetic orderings. Bosonization analysis is employed to give the concrete criterion for the stability of the helical edge states with interactions. We also performed the sign-problem-free quantum Monte-Carlo simulations which verified the interaction driven instabilities in the helical edge Luttinger liquids. For the Majorana edge modes of quantum wires, we study whether these zero modes can persist in an array of coupled wires, and if not, what their remnant might be. The bulk exhibits topologically distinct gapped phases and an intervening gapless phase. The interaction between Majorana zero modes and superfluid phases leads to spontaneous time-reversal symmetry breaking. Consequently, edge supercurrent loops emerge and edge Majorana fermions can be gapped out.

1) Congjun Wu, B. Andrei Bernevig, and Shou-Cheng Zhang, " The Helical Liquid and the Edge of Quantum Spin Hall Systems ",  Phys. Rev. Lett. 96 , 106401(2006).
2) Dong Zheng, Guang-Ming Zhang, Congjun Wu, "Particle-hole symmetry and interaction effects in the Kane-Mele-Hubbard model", Phys. Rev. B 84, 205121 (2011) .
3) Da Wang, Zhoushen Huang, and Congjun Wu, “Fate and remnants of Majorana zero modes in a quantum wire array”, Phys. Rev. B 89, 174510
4) Yi Li, Da Wang, Congjun Wu, “Spontaneous time-reversal symmetry breaking in the boundary Majorana flat bands,” New J. Phys. 15 085002 (2013).