You might ask yourself what makes \(\mathcal{T}^2=-1\) special, leading to the topological protection of the helical edge states.

As was mentioned in the first week, if \(\mathcal{T}^2=-1\) then Kramers’ theorem applies. Kramers’ theorem tells us that given an eigenstate \(|\Psi\rangle\) of the Hamiltonian with energy \(E\), its time-reversed partner \(|\Psi_\mathcal{T}\rangle\equiv\mathcal{T}|\Psi\rangle\) has the same energy, and the two states are orthogonal, \(\langle \Psi | \Psi_\mathcal{T}\rangle=0\). These two states form a so-called Kramers pair. As we already know, this leads to the fact that Hamiltonians with spinful time-reversal symmetry have two-fold degenerate energy levels - Kramers degeneracy.

Now, the two counterpropagating helical modes are time-reversed partners of each other, so they form precisely such a Kramers pair. The condition \(\langle \Psi | \Psi_\mathcal{T}\rangle=0\) implies that it is impossible to introduce any backscattering between the two states, unless we break time-reversal symmetry. This is the origin of the unit transmission and of the topological protection of helical edge states.

To gain a more intuitive understanding of this fact at a more microscopic level, we can assume that the projection of the electrons’ spin along a given axis is conserved, say the axis \(z\) perpendicular to the plane. Then at the edge you have, say, a right-moving mode with spin up and a left-moving mode with spin down, and no other modes if \(N=1\). Let’s draw again the picture of a helical edge state entering the disordered region:

Thus, an electron moving to the right must have spin up by assumption. In order to be reflected, its spin must also be flipped. However, this spin-flip scattering process is forbidden, and again we conclude that the electron is transmitted with probability one.

In the case \(\mathcal{T}^2=1\), there is no Kramers’ theorem. As a consequence, even though you can construct models which have counterpropagating edge states, you will find that they have no topological protection and can be gapped out without breaking the time-reversal symmetry.