Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport

We show that it is possible to have a topological phase in two-dimensional quasicrystals without any magnetic field applied, but instead introducing an artificial gauge field via dynamic modulation. This topological quasicrystal exhibits scatter-free unidirectional edge states that are extended along the system’s perimeter, contrary to the states of an ordinary quasicrystal system, which are characterized by power-law decay. We find that the spectrum of this Floquet topological quasicrystal exhibits a rich fractal (self-similar) structure of topological “minigaps,” manifesting an entirely new phenomenon: fractal topological systems. These topological minigaps form only when the system size is sufficiently large because their gapless edge states penetrate deep into the bulk. Hence, the topological structure emerges as a function of the system size, contrary to periodic systems where the topological phase can be completely characterized by the unit cell. We demonstrate the existence of this topological phase both by using a topological index (Bott index) and by studying the unidirectional transport of the gapless edge states and its robustness in the presence of defects. Our specific model is a Penrose lattice of helical optical waveguides—a photonic Floquet quasicrystal; however, we expect this new topological quasicrystal phase to be universal.

Popular Summary

Geometry and topology are fundamental aspects of physics; a particle can behave in completely different ways depending on the geometry of the underlying system. For example, quantum particles in a periodic lattice tend to be equally distributed in the whole lattice, whereas in a disordered lattice, they localize in a small region. Something more interesting happens in a quasicrystal—a structure that is not periodic but has a long-range order—where quantum particles are neither extended throughout the whole lattice nor localized, and the energy spectrum is fractal-like. Another example is the exotic class of materials called topological insulators, which are insulators in their interior but conductors along their edges or surfaces. Importantly, transport around the edges of a two-dimensional topological insulator is robust and unidirectional; it will not scatter to the bulk nor backscatter around the boundaries even in the presence of defects or disorder.

Topological insulators in periodic structures (lattices) have been observed for electrons, atoms, mechanical waves, and light. Would it be possible to create a topological insulator quasicrystal? In principle, in the absence of a magnetic field, no topological behavior is expected in quasicrystals; topological insulators must have at least one extended unidirectional edge state. On the other hand, the states of a quasicrystal are not extended and are immobile.

We show that a two-dimensional photonic quasicrystal—a photonic Penrose tiling lattice—exhibits a topological insulating phase with unidirectional edge states that extend around the perimeter of the quasicrystal and whose transport is robust regardless of the boundary shape. Importantly, we find an entirely new phenomenon: The fractal-like spectrum of the quasicrystal is endowed with topological band gaps at all energy scales; hence, new topological gaps appear as the system size increases. The fractal topological aspects discovered in this study lay the groundwork for new conceptual ideas involving topological phases in quasicrystals.

Figure 1

(a),(b) Periodic Penrose approximants of Penrose quasicrystals, containing 199 and 521 vertices. The red perimeter line defines the unit cell, which can be replicated periodically to fill a 2D plane, or—as we do here—folded to form a torus. (c) Sketch of the photonic Floquet quasicrystal formed by introducing a helical waveguide in each vertex of a Penrose tiling. (d) Nearest-neighbor distances $d_0<d_1<d_2$ used in the tight-binding model describing the quasicrystal.

Figure 2

(a) Quasienergy spectrum of a periodic Penrose tiling approximant containing 9349 vertices. ${\rho }_i$ mark the positions of the main gaps labeled (from the highest downwards) as $+t$, $+d$, $s$, $-d$, and $-t$. (b) Zoomed-in section of the quasienergy spectrum of quasiband $-d$. ${\sigma }_i$ mark the positions of the “subgaps” of the $-d$ band. (c) Left: Local patches associated with the ($+d,-d$) bands and the ($+t,-t$) bands, and local patches associated with the $-d$ band, giving rise to several subbands. Right: Sum of the modal intensities of the modes associated with the $-3$ subband.

Figure 3

Bulk states associated with each of the main bands $\{-t,-d,s,+d,+t\}$ of Fig. 2. Note the self-similar structure of the modes; i.e., they are formed by the repetition of a same pattern yet they do not extend over the entire lattice. Bottom right: A defect mode localized at one of the two mismatches of the Penrose approximants.

Figure 4

(a) Density of states of the (periodic) Penrose tiling approximant, as a function of the strength of the vector potential ${\mathrm{A}}_0$. (b) Zoomed-in section of the region marked by the frame in (a). The gaps enclosed by the red line are topological. (c) Zoomed-in section marked by the circle in (b), showing the transition from a topological gap to a nontopological (trivial) gap. As ${\mathrm{A}}_0$ is increased, the topological gap closes and a trivial gap opens up. (d) Energy spectrum around the topological gap for periodic approximants with increasing system size (red, orange, and yellow) and for a finite Penrose tiling (blue). The approximants are periodic; hence, they have a full band gap. On the other hand, the finite Penrose tiling has edge states whose energies reside in the topological gap.

Figure 5

Topological edge states associated with the topological band gap depicted in Fig. 4, for increasing size of the quasicrystal. The energy flow is always unidirectional as indicated by the blue lines.

Figure 6

Topological edge states associated with the topological gap shown in Fig. 4, for different shapes of the finite quasicrystal (a) circle, (b) arbitrary shape, (c) hole, and (d) a bulk defect (a missing waveguide in the bulk of the quasicrystal).

Figure 7

Propagation dynamics of a topological edge wave packet moving along the edges of the quasicrystalline lattice (with $A_0=1.75$), for (a) square and (b) arbitrary-shaped lattices, and (c) for a square hole in the lattice. Here, $z$ denotes the propagation distance in units of the period of the periodic modulation. The propagation is always unidirectional, with no reflections or scattering into the bulk, even for sharp edges and arbitrary shapes. Because the Chern number is $+1$, the propagation is counterclockwise around the outer edge, as in (a) and (b), but clockwise in a hole inside the quasicrystal, as in (c).

Figure 8

The topological features occur for several levels of resolution of the energy spectrum. The left-hand panels show three generations of the fractal energy spectrum (with $A_0=3$ and 15466 vertices), where the pink regions mark the topological band gaps. The right-hand panel shows the topological edge states associated with the second and third “generations” of the fractal spectrum of a finite topological quasicrystal, where (a)–(h) associate each edge state to each of the gaps shown in the left-hand panel.