Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder

The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number $\nu$ for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for $\nu$ and, using an explicit one-dimensional disordered topological model, we show that $\nu$ remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, $\nu$ remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of $\nu$ to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.

Figure 1

Evolution of the winding number $\nu$ [Eq. (6)] with disorder: $W_1=0.5W$ and $W_2=W$. The raw, unaveraged data for 200 disorder configurations are shown by the scattered points and the average by the solid line. Inset: The conduction and valence edges as functions of $W$, indicating a spectral gap closing at $W\approx 3$ (marked by the dashed line in the main panel). The marked data point reports a quantized $\nu =0.9994$, at disorder well beyond $W=3$.

Figure 2

(a) The critical surface ($\mathrm{\Lambda }\to \infty$) in the three-dimensional phase space ($m$, $W_1$, $W_2$). The lines (1) and (2) represent the singular points where the scaling is anomalous (see the text). The next panels report maps of the (b),(c) winding number and (d),(e) localization length as computed with Eq. (6) and with the numerical transfer matrix method, respectively, for two sections of the phase space defined by the constraints (b),(d) $W_2=2W_1=W$ and (c),(e) $m=0.5$. The analytic critical curves are shown as black and white lines in (b),(c) and (d),(e), respectively. The computations of $\nu$ were done for $N=1000$ and averaged over ten disorder configurations. The transfer matrix was iterated $10^8$ times.