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

Abstract

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.

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

“…And while the even Chern number in even dimension results from a pairing of the K 0 element specified by the Fermi projection with an even Fredholm module, hence leading to an even index theorem, the odd Chern number in odd dimension stems from the pairing of a K 1 element, the Fermi unitary operator U F , with an adequate odd Fredholm module, hence providing an odd index theorem. From a physical point of view, there are also major differences between the unitary and the chiral unitary classes, which were already revealed by the numerics of [17]. A non-trivial topological phase from class A necessarily posses extended states at some energies above and below the Fermi level, because the even Chern number of the Fermi projection vanishes when the Fermi level is sent into the low and high energy limits (hence a topological transition must occur in the process).…”

“…More recently, higher even Chern numbers have entered the theory of topological insulators [13,16,23] and an index theorem similar to Theorem 1.1 has been proved [22]. Also the odd Chern numbers have already appeared in the literature on topological insulators from the chiral unitry class (also called the AIIIsymmetry class) [17,26,28]. In fact, the ground state of a periodic system from this class can be uniquely characterized by a particular unitary matrix defined in the momentum-space (see Eq.…”

“…To present something concrete, let us write out the disordered one-dimensional model analyzed numerically in Ref. [17] …”

“…(7) below) and thus phases of such systems can be labelled by the odd Chern number (1). In [17], the position-space formula (3) was proposed as a phase label for disordered systems in the chiral unitary class. Furthermore, for explicit 1 and 3-dimensional topological models from the chiral unitary class, the invariant was evaluated numerically in the presence of strong disorder [17,27], by implementing techniques from [21].…”

“…In [17], the position-space formula (3) was proposed as a phase label for disordered systems in the chiral unitary class. Furthermore, for explicit 1 and 3-dimensional topological models from the chiral unitary class, the invariant was evaluated numerically in the presence of strong disorder [17,27], by implementing techniques from [21]. It was found that Ch d (U ) remains quantized and non-fluctuating as the disorder strength is increased, up to a critical disorder strength where the localization length of the system diverges and the invariant sharply changes its value.…”