The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν 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 ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν 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.

The Weyl semimetal (WSM) is a newly proposed quantum state of matter. It has Weyl nodes in bulk excitations and Fermi arcs surface states. We study the effects of disorder and localization in WSMs and find three exotic phase transitions. (I) Two Weyl nodes near the Brillouin zone boundary can be annihilated pairwise by disorder scattering, resulting in the opening of a topologically nontrivial gap and a transition from a WSM to a three-dimensional (3D) quantum anomalous Hall state. (II) When the two Weyl nodes are well separated in momentum space, the emergent bulk extended states can give rise to a direct transition from a WSM to a 3D diffusive anomalous Hall metal. (III) Two Weyl nodes can emerge near the zone center when an insulating gap closes with increasing disorder, enabling a direct transition from a normal band insulator to a WSM. We determine the phase diagram by numerically computing the localization length and the Hall conductivity, and propose that the exotic phase transitions can be realized on a photonic lattice.PACS numbers: 72.15. Rn, 73.20.Fz, Topological quantum states of matter have emerged as an important and growing field in condensed matter and materials physics recently [1,2]. The Weyl semimetal (WSM) is a newly proposed quantum state of the kind that breaks time-reversal symmetry or inversion symmetry [3][4][5][6][7][8][9][10]. A WSM exhibits a set of paired zero-energy Weyl nodes (linearly touching points of conduction and valence bands) in its bulk spectrum and Fermi arcs excitations localized on the surface. A number of candidate materials have been predicted to be WSMs, including pyrochlore iridates and magnetic topological insulator multilayers [3][4][5][6]. Recently, following the theoretical prediction [7], angle resolved photoemission experiments confirmed that TaAs is a WSM by the observation of the Fermi arcs surface states [8]. Both the Weyl nodes and the Fermi arcs have been observed in NbAs using a combination of soft X-ray and ultraviolet photoemission experiments [9]. Furthermore, the Weyl points have been predicted and subsequently observed remarkably in gyroid photonic crystals [10].In this Letter, we study both numerically and analytically the stability of the gapless Weyl nodes and Fermi arcs against random potential scattering and the novel disorder-induced metal-insulator transitions in WSM systems. Previous studies have concentrated on the properties of a single Weyl node, assuming that the disorder potential is smooth enough to avoid scattering between different nodes [11][12][13][14]. Indeed, a system with a single Weyl node is not subject to Anderson localization even for strong disorder [15]. However, the theorem of Nielsen and Ninomiya states that gapless Weyl nodes with opposite chirality must appear in pairs [16]. Thus, it is essential to study the localization properties of a pair of Weyl nodes since they can be annihilated pairwise when approaching each other in momentum space or by strong intervalley scattering [17,18]. To this end, we study a model syste...

Using an explicit 1-dimensional model, we provide direct evidence that the one-dimensional topological phases from the AIII and BDI symmetry classes follow a Z-classification, even in the strong disorder regime when the Fermi level is embedded in a dense localized spectrum. The main tool for our analysis is the winding number ν, in the non-commutative formulation introduced in I. Mondragon-Shem, J. Song, T. L. Hughes, and E. Prodan, arXiv:1311.5233. For both classes, by varying the parameters of the model and/or the disorder strength, a cascade of sharp topological transitions ν = 0 → ν = 1 → ν = 2 is generated, in the regime where the insulating gap is completely filled with the localized spectrum. We demonstrate that each topological transition is accompanied by an Anderson localization-delocalization transition. Furthermore, to explicitly rule out a Z 2 classification, a topological transition between ν = 0 and ν = 2 is generated. These two phases are also found to be separated by an Anderson localization-delocalization transition, hence proving their distinct identity.

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