Bulk Localised Transport States in Infinite and Finite Quasicrystals via Magnetic Aperiodicity

Abstract

Robust edge transport can occur when charged particles in crystalline lattices interact with an applied external magnetic field. This system is well described by Bloch's theorem, with the spectrum being composed of bands of bulk states and in-gap edge states. When the confining lattice geometry is altered to be quasicrystaline, i.e. quasiperiodic, then Bloch's theorem breaks down. However, for the quasicrystalline system, we still expect to observe the basic characteristics of bulk states and current carrying edge states. Here, we show that for quasicrystals in magnetic fields, there is also a third option; the bulk localised transport states. These states share the in-gap nature of the well-known edge states and can support transport along them, but they are fully contained within the bulk of the system, with no support along the edge. This results in transport being possible both along the edge and within distinct regions of the bulk. We consider both finite-size and infinite-size systems, using rigorous error controlled computational techniques that are not prone to finite-size effects. The bulk localised transport states are preserved for infinite-size systems, in stark contrast to the normal edge states. This allows for transport to be observed in infinite-size systems, without any perturbations, defects, or boundaries being introduced. We confirm the ingap topological nature of the bulk localised transport states for finite and infinite-size systems by computing common topological measures; namely the Bott index and local Chern marker. The bulk localised transport states form due to a magnetic aperiodicity arising from the interplay of length scales between the magnetic field and quasiperiodic lattice. Bulk localised transport could have interesting applications similar to those of the edge states on the boundary, but that could now take advantage of the larger bulk of the lattice. The infinite size techniques introduced here, especially the calculation of topological measures, could also be widely applied to other crystalline, quasicrystalline, and disordered models.

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

“…With this technique in hand, we can reliably probe the spectral properties of systems in infinite dimensions. Indeed, this technique is already allowing for the discovery and investigation of new physics in quasicrystalline systems, including their transport and topological properties [65].…”

1D quasicrystals and topological markers

“…One of the most prevalent topological markers is the Chern marker which is defined for 2D time independent systems and has previously been used to investigate the topological structure of inhomogeneous systems [13][14][15][16][17][18][19][20][21][22]. The Chern marker has also been used to investigate the topological properties of 2D quasicrystals with different aperiodic tiling structures [23,24]. While the Chern marker is defined for 2D systems it was recently shown by the authors of this paper that a topological marker can be defined for 1D time-dependent systems and be used to determine their topological index [25].…”

A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations

“…Contour methods can also be interpreted in terms of rational approximations[46]. For another rational function approach to TFDEs, see[47].2 This "solve-then-discretise" paradigm has recently been applied to spectral computations[50][51][52][53][54][55], extensions of classical methods such as the QL and QR algorithms[56,57] (see also[58]), Krylov methods[59,60], spectral measures[61- 63] and computing semigroups[64]. Related work includes that of Olver, Townsend and Webb, providing a foundational and practical framework for infinite-dimensional numerical linear algebra and computations with infinite data structures[65][66][67][68].3 See Theorem 3.5 for our specific example.…”