Speaking of the scaling flow: the scaling hypothesis appears to be very powerful, does the topology have any impact on it?
We can understand some limits. First of all, if \(g\gg 1\), the system is a metal, and there can hardly be any trace of topology left. However when \(g \lesssim 1\), the system can become one out of several topologically distinct types of insulators, and we can expect some sort of transition between different insulating phases.
The rational assumption at this point is to assume that the scaling hypothesis still holds, but in addition to \(g\) it depends on the average topological invariant of the disordered ensemble \(\langle \mathcal{Q} \rangle\).
Just like in the case of the non-topological phases, there is no universal proof that this form of scaling flow is correct. Instead, there is a vast amount of mixed numerical and analytical evidence that this is correct.
Let’s try and verify our hypothesis by constructing the scaling flow of the disordered Kitaev chain. We can do it in the following way:
Choose a given disorder strength \(U\), a given length \(L\), and a set of values \(\{\mu\}\) for the chemical potential. For each of these values, we compute \(\langle \mathcal{Q} \rangle\) and the average transmission \(\langle T \rangle\) over a large number of disorder realizations (as large as possible, in our case 1000). This gives us a set of starting points on the \((\langle \mathcal{Q} \rangle, \langle T \rangle)\) plane.
Increase the chain length \(L\), and compute \(\langle \mathcal{Q} \rangle, \langle T \rangle\) again for all the values \(\{\mu\}\). We obtain a second set of points on the plane.
Join the points corresponding to the same value of \(\mu\).
Increase again the length \(L\), and so on…
Here’s what we get:
The lines have a direction, which tells us how \(\langle Q \rangle\) and \(\langle T \rangle\) change as we increase \(L\). In the plot above, \(L\) is increasing in going from bright to dark colors.
The first and the most important observation we can make is that the lines do not intersect, which confirms the scaling hypothesis.
Most of the lines tend to one of the two points \((\langle \mathcal{Q} \rangle, \langle T \rangle) = (-1, 0)\) or \((1, 0)\). These correspond to two insulating systems with different topological invariants. We can say that quantized values of \(\langle \mathcal{Q} \rangle\) are insulating, and they correspond to attractive fixed points of the flow.
You can also see that the flow is roughly separated around a vertical critical line at \(\langle \mathcal{Q} \rangle=0\). All lines which start from a negative value of \(\langle \mathcal{Q} \rangle\) end up at \((-1, 0)\), and all the lines which start from a positive value end up at \((1,0)\).
Finally, the point \((0, 0)\) is a saddle point: the flow goes towards it along the vertical axis, and away from it along the horizontal axis.
We now understand better why two parameter scaling is necessary in the presence of a topological invariant. Even in the presence of disorder, there can be topologically distinct insulating phases. Therefore, saying that every system flows to an insulator is not enough anymore. Including the average topological invariant as a second scaling parameter allows to predict towards which insulating phase the system will flow.
The flow, we just calculated is in fact valid for all one-dimensional topological insulators and superconductors. In the case of a \(\mathbb{Z}\) invariant, the saddle points are located at \((\langle \mathcal{Q} \rangle, \langle T\rangle) = (n+1/2, 0)\).
It is important to notice that one important result of the standard scaling theory regarding one dimensional system remains true: in the plot above all lines flow to no transmission, or in other words there are no metallic phases in the flow diagram.