Quasiperiodic tilings in a magnetic field

Vidal, Julien; Mosseri, Rémy

doi:10.1016/j.jnoncrysol.2003.11.027

Cited by 12 publications

(18 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Instead, we show that the 2D quasicrystal shares the topological properties of a 2D Chern insulator, namely, it is characterized by non-zero Chern numbers and chiral edge states at its 1D boundary. Our results provide a novel interpretation for the Hofstadter butterfly-like spectrum associated with the quasicrystal [26]: by analyzing the topological order of the system, we show that this fractal energy spectrum hosts a variety of Chern (Hofstadter) insulators [27]. Contrary to perturbative approaches aiming to study the robustness of quantum Hall phases in the presence of weak disorder, the Chern insulating phases revealed in this work are associated with a genuine non-periodic 2D system (i.e.…”

Section: Introductionmentioning

confidence: 66%

“…For the sake of completeness, we now sketch the cut-and-project method in the following Section II A, see also Refs. [25,26]. This Section also introduces the isometric generalized Rauzy tiling (iGRT) [26], where all the nearest-neighboring links share the same length.…”

Section: The Modelmentioning

confidence: 99%

“…[25,26]. This Section also introduces the isometric generalized Rauzy tiling (iGRT) [26], where all the nearest-neighboring links share the same length. In this setting, all the tiles of the quasi-periodic tiling also have the same area [ Fig.…”

Section: The Modelmentioning

confidence: 99%

See 2 more Smart Citations

Topological Hofstadter insulators in a two-dimensional quasicrystal

et al. 2015

View full text Add to dashboard Cite

We investigate the properties of a two-dimensional quasicrystal in the presence of a uniform magnetic field. In this configuration, the density of states (DOS) displays a Hofstadter butterfly-like structure when it is represented as a function of the magnetic flux per tile. We show that the low-DOS regions of the energy spectrum are associated with chiral edge states, in direct analogy with the Chern insulators realized with periodic lattices. We establish the topological nature of the edge states by computing the topological Chern number associated with the bulk of the quasicrystal. This topological characterization of the non-periodic lattice is achieved through a local (real-space) topological marker. This work opens a route for the exploration of topological insulating materials in a wide range of non-periodic lattice systems, including photonic crystals and cold atoms in optical lattices.

show abstract

Section: Introductionmentioning

confidence: 66%

Section: The Modelmentioning

confidence: 99%

See 1 more Smart Citation

Topological Hofstadter insulators in a two-dimensional quasicrystal

et al. 2015

View full text Add to dashboard Cite

show abstract

“…[11]. Its commensurate version (also known as the isometric Rauzy tiling) is obtained by modifying the projection direction [15]. Edges in the Z 3 lattice -as represented by three canonical basis vectors -are transformed into edges in the tiling plane R 2 as follows:…”

Section: Commensurate Rauzy Tiling and Underlying Triangular Latticementioning

confidence: 99%

Landau levels in quasicrystals

2018

Self Cite

View full text Add to dashboard Cite

Two-dimensional tight-binding models for quasicrystals made of plaquettes with commensurate areas are considered. Their energy spectrum is computed as a function of an applied perpendicular magnetic field. Landau levels are found to emerge near band edges in the zero-field limit. Their existence is related to an effective zero-field dispersion relation valid in the continuum limit. For quasicrystals studied here, an underlying periodic crystal exists and provides a natural interpretation to this dispersion relation. In addition to the slope (effective mass) of Landau levels, we also study their width as a function of the magnetic flux and identify two fundamental broadening mechanisms: (i) tunneling between closed cyclotron orbits and (ii) individual energy displacement of states within a Landau level. Interestingly, the typical broadening of the Landau levels is found to behave algebraically with the magnetic field with a nonuniversal exponent. PACS numbers: 71.23.Ft,71.70.Di,73.43.-f 1 These commensurate quasicrystals have perfect long-range order and are no less quasicrystalline than the original ones. The adjective "commensurate" only refers to the fact that the ratios of their plaquette areas are rational numbers. arXiv:1807.10032v2 [cond-mat.dis-nn]

show abstract

“…Second, as for any system, if one considers open boundary conditions, edge states prevent one from identifying bulk gaps properly as discussed in Refs. [16,17]. Here, we solve these two issues by: (i) deforming the tiling to deal with identical tile areas (see discussion above) and (ii) by considering periodic boundary conditions.…”

Section: Boundary Conditions and Gauge Choicementioning

confidence: 99%

Hofstadter butterfly of a quasicrystal

Fuchs

Vidal

2016

Phys. Rev. B

Self Cite

View full text Add to dashboard Cite

The energy spectrum of a tight-binding Hamiltonian is studied for the two-dimensional quasiperiodic Rauzy tiling in a perpendicular magnetic field. This spectrum known as a Hofstadter butterfly displays a very rich pattern of bulk gaps that are labeled by four integers, instead of two for periodic systems. The role of phason-flip disorder is also investigated in order to extract genuinely quasiperiodic properties. This geometric disorder is found to only preserve main quantum Hall gaps.

show abstract

Quasiperiodic tilings in a magnetic field

Cited by 12 publications

References 34 publications

Topological Hofstadter insulators in a two-dimensional quasicrystal

Topological Hofstadter insulators in a two-dimensional quasicrystal

Landau levels in quasicrystals

Hofstadter butterfly of a quasicrystal

Contact Info

Product

Resources

About