Topological equivalence of crystal and quasicrystal band structures

TL;DR

This paper demonstrates that topologically non-trivial states in one-dimensional incommensurate lattice models are fundamentally equivalent to those in commensurate models, challenging the idea that incommensurability introduces new topological classifications.

Contribution

The authors provide an explicit construction showing incommensurability does not alter the topological classification, establishing a continuous connection between commensurate and incommensurate states without gap closing.

Findings

End states in incommensurate models originate from the same mechanisms as in commensurate models.

Incommensurability does not provide a meaningful connection to higher-dimensional topological classifications.

States can be smoothly interpolated between commensurate and incommensurate without closing the band gap.

Abstract

A number of recent articles have reported the existence of topologically non-trivial states and associated end states in one-dimensional incommensurate lattice models that would usually only be expected in higher dimensions. Using an explicit construction, we here argue that the end states have precisely the same origin as their counterparts in commensurate models and that incommensurability does not in fact provide a meaningful connection to the topological classification of systems in higher dimensions. In particular, we show that it is possible to smoothly interpolate between states with commensurate and incommensurate modulation parameters without closing the band gap and without states crossing the band gap.