  We have found the first instance of a third-order topological Anderson
insulator (TOTAI). This disorder-induced topological phase is gapped and
characterized by a quantized octupole moment and topologically protected corner
states, as revealed by a detailed numerically exact analysis. We also find that
the disorder-induced transition into the TOTAI phase can be analytically
captured with remarkable accuracy using the self-consistent Born approximation.
For a larger disorder strength, the TOTAI undergoes a transition to a trivial
diffusive metal, that in turn becomes an Anderson insulator at even larger
disorder. Our findings show that disorder can induce third-order topological
phases in 3D, therefore extending the class of known higher-order topological
Anderson insulators.
