Quantum criticality of the topological Anderson insulator
<br>
<br>
In the presence of even weak amounts of disorder, low dimensional topological
band insulators turn into topological Anderson insulators (tAI). Translational
invariance being absent, the tAI must be described by  concepts different from
the clean limit band structures classification schemes. In this talk we argue
that much of the universal physics of the tAI is contained in the system size
dependent flow of two parameters, the first of which is an average transport
coefficient (at any finite size, the tAI is a conductor), and the second the
mean value of a now statistically distributed topological index. These two
parameters exhibit flow similar to that of the Pruisken-Khmelnitskii flow
diagram of the quantum Hall insulator. Specifically, the flow describes
describes quantum criticality at topological phase transitions, the approach
towards an insulating configuration away from criticality, and, along with it,
the emergenece of a self averaging integer index. For some symmetry classes,
that flow can be established in closed analytic form. However, we argue that the
overall picture is of more general validity and provides a unified framework to
describe both the bulk and the surface physics of the topological Anderson
insulator.