The Majorana zero mode is the consequence of a topological phase in the topological superconductor, and its presence is dictated by the bulk-boundary correspondence. Can we find any consequence of this fact in \(r_0\)? It turns out that reflection matrices \(r\) with particle-hole symmetry are also topological in their own way. Their topological invariant is

\[Q = \det\,r_0\,.\]

Again, we will not derive this equation, but rather convince ourselves this expression is correct.

First of all, the determinant of a unitary matrix such as \(r_0\) is always a complex number with unit norm, so \(\left|\det\,r_0\,\right|=1\). Second, because of particle-hole symmetry, the determinant is real: \(\det r_0 = \det\, (\tau_x r^*_0\,\tau_x) = \det\,r_0^*\,=(\det\,r_0)^*\). Hence, \(\det\,r_0\,= \pm 1\). This is quite promising! Two possible discrete values, just like the Pfaffian invariant of the Kitaev chain.

Because it is just dictated by unitarity and particle-hole symmetry, the determinant of \(r_0\) cannot change from \(+1\) to \(-1\) under a change of the properties of the NS interface. For instance, you can vary the height of the potential barrier at the interface, but this cannot affect the determinant of \(r_0\).

The only way to make the determinant change sign is to close the bulk gap in the superconducting electrode. If the gap goes to zero, then it is not true that an incoming electron coming from the normal metal can only be normal-reflected or Andreev-reflected. It can also just enter the superconducting electrode as an electron. Hence the reflection matrix no longer contains all the possible processes taking place at the interface, and it won’t be unitary anymore. This allows the determinant to change sign. We conclude that \(Q=\det\,r\) is a good topological invariant.

Explicitly, we have that

\[Q=|r_{ee}|^2-|r_{eh}|^2\equiv\pm 1\,.\]

We already saw that unitarity requires that \(|r_{ee}|^2+|r_{eh}|^2=1\). There are only two possibilities for both conditions to be true: either \(|r_{ee}|=1\) (perfect normal reflection) or \(|r_{eh}|=1\) (perfect Andreev reflection). The situation cannot change without a phase transition. Thus the quantized conductance of the Majorana mode is topologically robust in this case, and in fact survives past the tunneling limit.