We study the topological gap labeling of general 3D quasicrystals and we find that every gap in the spectrum is characterized by a set of the third Chern numbers. We show that a quasi-periodic structure has multiple Brillouin zones defined by redundant wavevectors, and the number of states below a gap is quantized as an integer linear combination of volumes of these Brillouin zones. The associated quantum numbers to characterize energy gaps can be expressed as third Chern numbers by considering a formal relationship between an adiabatic charge pumping under cyclic deformation of the quasi-periodic potential and a topological nonlinear electromagnetic response in 6D insulators.
Chern insulator or quantum anomalous Hall state is a topological state with integer Hall conductivity but in absence of Landau level. It had been well established on various two-dimensional lattices with periodic structure. Here, we report similar Chern insulators can also be realized on the quasicrystal with $5$-fold rotational symmetry. Providing the staggered flux through plaquettes, we propose two types of quasicrystalline Chern insulators. Their topological characterizations are well identified by the robustness of edge states, non-zero real-space Chern number, and quantized conductance. …
Topological physics in photonic systems have attracted great attentions in recent years. In this work, we theoretically study the one and two dimensional photonic quasicrystal resonator lattices characterized by the first and second Chern number, which show exotic boundary states within the photonic energy band gap. In particular, the second Chern number protected edge states has opened up new possibilities for realizing topological physics of dimensions higher than three in photonic systems, which is highly sought for. Such photonic systems can be easily experimentally realized in regular pho…
We report on a study of topological properties of Fibonacci quasicrystals. Chern numbers which label the dense set of spectral gaps, are shown to be related to the underlying palindromic symmetry. Topological and spectral features are related to the two independent phases of the scattering matrix: the total phase shift describing the frequency spectrum and the chiral phase sensitive to topological features. Conveniently designed gap modes with spectral properties directly related to the Chern numbers allow to scan these phases. An effective topological Fabry-Perot cavity is presented.
The topological properties of the quantum Hall effect in a crystalline lattice, described by Chern numbers of the Hofstadter butterfly quantum phase diagram, are deduced by using a geometrical method to generate the structure of quasicrystals: the cut and projection method. Based on this, we provide a geometric unified approach to the Hofstadter topological phase diagram at all fluxes. Then we show that for any flux, the bands conductance follow a two letter symbolic sequence . As a result, bands conductance at different fluxes obey inflation/deflation rules as the ones observed to build quasi…
We study the role of the edges in determining the features of the topological phase in a quasicrystalline higher order topological insulator. We consider a specific model consisting of two stacked Haldane models with opposite Chern number and a $30^\circ$ twist, whose structure is crystallographically equivalent to that of the graphene quasicrystal. We find that the gap-opening in the low energy spectrum of the higher order topological insulator occurs at different energies when different kinds of edges are considered. Crucially, bearded bonds appear to be necessary for the gap to appear close…
The bulk spectrum of a possible Chern insulator on a quasicrystalline lattice is examined. The effect of being a 2D insulator seems to override any fractal properties in the spectrum. We compute that the spectrum is either two continuous bands, or that any gaps other than the main gap are small. After making estimates on the spectrum, we deduce a finite system size, above which the K-theory must coincide with the K-theory of the infinite system. Knowledge of the spectrum and $K$-theory of the infinite-area system will control the spectrum and K-theory of sufficiently large finite systems. The …
We study the topological characterization of the energy gaps in general two-dimensional quasiperiodic systems consisting of multiple periodicities, represented by twisted two-dimensional materials. We show that every single gap is uniquely characterized by a set of integers, which quantize the area of the momentum space in units of multiple Brillouin zones defined in the redundant periodicities. These integers can be expressed as the second Chern numbers, by considering an adiabatic charge pumping under a relative slide of different periodicities, and using a formal relationship to the four-di…
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 topo…
We investigate the topological properties of Fibonacci quasicrystals using cavity polaritons. Composite structures made of the concatenation of two Fibonacci sequences allow investigating generalized edge states forming in the gaps of the fractal energy spectrum. We employ these generalized edge states to determine the topological invariants of the quasicrystal. When varying a structural degree of freedom (phason) of the Fibonacci sequence, the edge states spectrally traverse the gaps, while their spatial symmetry switches: the periodicity of this spectral and spatial evolution yields direct m…
Explicit equivalence between the spectral localizer and local Chern and winding markers
KTH Royal Institute of Technology（皇家理工学院）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基克科学基金会）；Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、内耳研究所）
Nature harbors crystals of dimensionality ($d$) only up to three. Here we introduce the notion of \emph{projected topological branes} (PTBs): Lower-dimensional branes embedded in higher-dimensional parent topological crystals, constructed via a geometric cut-and-project procedure on the Hilbert space of the parent lattice Hamiltonian. When such a brane is inclined at a rational or an irrational slope, either a new lattice periodicity or a quasicrystal emerges. The latter gives birth to topoquasicrystals within the landscape of PTBs. As such PTBs are shown to inherit the hallmarks, such as the …
Aharonov-Bohm caging is a localization mechanism stemming from the competition between the geometry and the magnetic field. Originally described for a tight-binding model in the dice lattice, this destructive interference phenomenon prevents any wavepacket spreading away from a strictly confined region. Accordingly, for the peculiar values of the field responsible for this effect, the energy spectrum consists of a discrete set of highly degenerate flat bands. In the present work, we show that Aharonov-Bohm cages are also found in an infinite set of hyperbolic dice tilings defined on a negative…
