I have a spin 1/2 chain with open boundary conditions described by Hamiltonian $H=\sum_i \sigma_{2i}^z \sigma_{2i+1}^z$. From $H$ it's clear that boundary sites are decoupled from the rest of the system. Thus, there are free spin 1/2s at the edges and G.S. will be 4-fold degenerate. These edge modes will be destroyed if I add a perturbation which breaks time-reversal symmetry (TRS) (e.g. If we add a magnetic field then spins will be polarized and degeneracy will be lifted). Its clear that it's a topological phase but how can it be shown rigorously that if we add any perturbation respecting TRS, edge modes will survive?
Is there a way to show that through the projective representation of TRS group. Since these free spins will transform in a non-trivial way under TRS ($\mathcal(T)^2=-1$).