Geometry and topology tango in ordered and amorphous chiral matter

Abstract

Systems as diverse as mechanical structures and photonic metamaterials enjoy a common geometrical feature: a sublattice or chiral symmetry first introduced to characterize electronic insulators. We show how a real-space observable, the chiral polarization, distinguishes chiral insulators from one another and resolve long-standing ambiguities in the very concept of their bulk-boundary correspondence. We use it to lay out generic geometrical rules to engineer topologically distinct phases, and design zero-energy topological boundary modes in both crystalline and amorphous metamaterials.

Topology Classification using Chiral Symmetry and Spin Correlations in Graphene Nanoribbons

“…After this paper was written and submitted to the arXiv, we became aware of another work on the arXiv that also uses chiral symmetry to classify 1D bipartite models.…”

Estimating bulk and edge topological indices in finite open chiral chains

“…where n A,θ − n B,θ counts the difference of number of A/B site in the region where θ = 1. It can be interpreted as a chiral polarization of the sites in the support of θ and implies that the number of edge modes do not entirely depend on pure-bulk properties, as already pointed in [17,12]. This extra term is H-independent and a pure lattice property, therefore it is still easy to compute.…”

“…Thus, the edge index is the chiral density of low energy states integrated in the left part of the chain. If the bulk Hamiltonian has a gap (−∆, ∆) and δ ∆, the edge index counts the polarization of the edge modes near zero energy, and localized in the left part of the chain [17,12]. Theorem 1.…”

Amorphous topological matter: Theory and experiment

“…Among the different methods to characterize topological phases far from translationally invariant limits topological markers are a wide-spread tool. Topological marker is a unifying term that includes the local markers [49][50][51][52][53][54][55][56], the spectral localizers [57][58][59][60], the nonlocal (spin) Bott indices [46,[61][62][63][64][65][66][67][68][69], and similar generalizations of the winding of the quadrupole and octupole moment [42,43]. Markers characterizing the two-dimensional quantum Hall phase [70,71] are especially well explored, including the local Chern marker [49,50] and the nonlocal Bott index [63].…”