Abstract
We present Floquet fractal topological insulators: photonic topological insulators in a fractaldimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivialtotopological phase transition. The quasienergy spectrum shows the existence of topological edge states corresponding to realspace Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid halffractal, halfhoneycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics.
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Introduction
Topological insulators are a new phase of matter characterized by an insulating bulk and perfectly conductive edges^{1,2}. They have been at the forefront of condensed matter physics for the past decade and more recently inspired the emergence of topological phases in many classicalwave systems^{3,4,5}, such as microwaves^{6,7,8}, photonics^{3,9,10,11,12,13,14,15,16}, acoustics^{4}, and more. Photonics specifically has become the cuttingedge platform for exploring all kinds of topological phases ranging from the quantum spin Hall effect^{12}, Floquet topological insulators^{13}, topological crystalline insulator^{16}, and valley Hall effect^{17,18}; all the way to topological systems that lack periodicity, such as topological quasicrystals^{19} and even topological Anderson insulators, in which the topology is induced by disorder^{20}. Thus far, all studies of topological insulators have explored systems in integer dimensions (physically, 2D or 3D) with a welldefined bulk and edges. However, the physical dimensions do not always define the dimensions in which a system evolves: some structures have a noninteger (fractal) dimension, despite being in a 2D or 3D realm. The existence of systems with fractal dimensions raises a series of fascinating questions in the context of topological physics. For example, is it possible to realize topological edge states in fractal dimensions? Moreover, fractal structures tend to include holes, so can topological edge states be found around every hole in the system or only in the external boundary? Intuitively, one might think that there are no topological edge states because our fractal lattices often contain no bulk at all, hence one cannot rely on the bulkedge correspondence^{21,22} to predict topological edge states in fractal lattices. This raises a deeper question: is there bulkedge correspondence when the fractal structure is actually made up of holes within the bulk?
Here, we investigate the photonic Floquet topological phase in a periodically driven fractal lattice. This lattice relies on a fractal photonic crystal [the Sierpinski gasket (SG)] consisting of evanescently coupled helical waveguides, which can be realized by femtosecondlaserwriting technology^{23}. We calculate the topological Floquet spectrum and show the existence of topological edge states corresponding to realspace Chern number 1^{24,25}, which can be controlled by periodic driving. We explore the dynamics of the edge states and their robustness in simulations in the fractal SG lattice and find that wavepackets made up of topological edge states propagate along the outer edge without penetration into the bulk and without backscattering even in the presence of disorder and sharp corners. Likewise, the topological edge states associated with inner edges in the fractal lattice exhibit robust transport whenever the inner edge includes a large enough area. These results imply that fractal structures can act as topological insulators, despite the lack of periodicity and the structures being made up mostly of holes. Subsequently, we study transport in a hybrid lattice combining the fractal lattice with a honeycomb lattice and find that topological edge states can pass from the honeycomb lattice into the edge of the fractal lattice and vice versa, where they exhibit topologically protected transport. This observation further demonstrate that the edge states in the fractal lattice directly correspond to the same Chern number as that of a honeycomb lattice driven by the same periodic modulation. Finally, it is possible to obtain similar results with other fractal platforms: the Sierpinski carpet under an aperiodic arrangement, and we conjecture that the 3D realizations of both the SG and the Sierpinski carpet also give rise to topological edge states and, likewise, the Cantor cubes and Cantor dust. Hence, our results suggest a wealth of new kinds of topological systems and new applications, such as using topological robustness combined with the enhanced sensitivity of fractal systems for sensing and, in nonHermitian settings, topological insulator lasers^{26,27,28} in fractal dimensions.
Results
Our starting point is the SG with a Hausdorff dimension \(d_f = {\mathrm{ln}}\left( 3 \right)/{\mathrm{ln}}\left( 2 \right) = 1.585\). Consider a photonic lattice of evanescently coupled helical waveguides, similar to ref. ^{13}. Figure 1 shows the iterative generations of the SG. As can be seen, the first generation G(1) of the SG has nine blue circles, which indicate the positions of the helical waveguides. Generation G(2) consists of three copies of G(1), sharing three vertices. Accordingly, the G(2) waveguide lattice has 24 waveguides organized as the second generation of the SG. Similarly, G(n) has three copies of G(n − 1), sharing three corner sites. Hereafter, we focus on fractal lattices of generations G(4) and G(5), and we conjecture that the conclusions we draw from this study hold for the SG lattice in any generation. Examining Fig. 1 reveals that all sites in the SG fractal lattice are on the boundaries, and there is not even a single site that does not reside on a boundary—external or internal. Finally, as in ref. ^{13}, this lattice consists of helical waveguides, which is equivalent to a periodically driven potential that introduces an artificial gauge field \({\boldsymbol{A}}\).
The equation governing the diffraction of light in this fractal photonic lattice under the tightbinding approximation^{13} can be written as
where z is the optical axis, \(\psi _n\) is the amplitude of the electric field in the nth waveguide, \(c_0\) is the coupling strength, \({\boldsymbol{r}}_{m,n}\) is the displacement vector pointing from waveguide m to waveguide n, \({\boldsymbol{A}}\left( z \right) = A_0\left[ {\sin \left( {{\mathrm{\Omega }}z} \right),  \cos \left( {{\mathrm{\Omega }}z} \right),0} \right]\) is the artificial vector potential induced by the helicity of the waveguides with amplitude \(A_0 = kR{\mathrm{\Omega }}\), in which \(k\) is the wavenumber of the light in the medium, \(R\) is the radius of the helix, \({\mathrm{\Omega }}\) is the longitudinal frequency of the helix corresponding to periodicity \(L = 2\pi /{\mathrm{\Omega }}\), and m indicates that the summation is taken over all the nearest waveguides to waveguide n. The light evolution in the system is described by the paraxial wave equation, which is mathematically equivalent to the Schrödinger equation, with the zaxis playing the role of time. Equation (1) is derived by applying the tightbinding approximation to the paraxial wave equation.
The eigenvalues and eigenstates can be obtained by diagonalizing the unitary evolution operator for one period^{29}. The results of the quasienergy spectrum \(\beta\) (which in a photonic lattice are the deviation of the propagation constant from the wavenumber in the medium^{13}) in the fractal SG systems are shown in Fig. 2a, with \({\boldsymbol{A}}\left( z \right) = 0\) corresponding to the straight waveguides, and in Fig. 2b, with \({\boldsymbol{A}}\left( z \right)\,\ne\, 0\) corresponding to the helical waveguides. The spectrum for our G(4) fractal lattice is organized into five bunches (“bands”) separated by gaps (gray shaded regions in Fig. 2a, b). The spectrum of the nontopological system (Fig. 2a) shows a large central gap, with a flat “band” in the midgap. These states are immobile and degenerate (they all have the same energy), as expected from a nontopological system. On the other hand, for the driven system (the helical waveguides), as shown in Fig. 2b, the edge states from the central flat band evolve into nondegenerate unidirectional edge states. Figure 2c shows the field intensities of the actual wavefunctions of these eigenstates, specifically states 93 and 95 (out of 204 eigenstates), with quasienergies of −0.040 and −0.018, respectively. These states are localized at the exterior (state number 95) and the interior (state number 93) edges. As we show below, these states behave as topological edge states, exhibiting Chern number 1 and topologically protected transport. Note that in other bunches with quasienergies below −0.2 or above 0.2, the eigenstates are bulk states.
To verify that the edge states we have found [the nondegenerate unidirectional states in the rectangle of Fig. 2b, two of which are shown in Fig. 2c] are indeed topological, we need to characterize our system through its Chern number. Since fractal lattices are nonperiodic, we calculate the realspace Chern number^{24,25}. Heuristically, the realspace Chern number “measures” the chirality of states at a specific quasienergy, and in periodic systems, it yields the same integer number as the “standard” Chern number (defined on the momentum space)^{24,25}. The definition of the realspace Chern number is
where \(j,k,l\) are the lattice site indices within three different neighboring regions A–C [as drawn in the inset of Fig. 3b, arranged anticlockwise], \(P_{jk} = \langle{j{\mathrm{}}P{\mathrm{}}k}\rangle\) and the projector operator \(P\) projects onto a given state of a specific quasienergy (a state with quasienergy playing the role of the Fermi level). The results are shown in Fig. 3a, b. We calculate the realspace Chern number for our fractal lattice and, for a direct comparison, also for a honeycomb lattice, with both being driven by the same periodic modulation (manifested here as the helicity of the waveguides). The lower panels in Fig. 3 show the methodology of the calculation: the hexagons are divided into three distinct regions (A–C), each enclosing many helical waveguides, for both the fractal and honeycomb lattices. As expected, the helicity induces a topological bandgap in the honeycomb lattice [Fig. 3a] corresponding to realspace Chern number 1, which coincides with the outcome of the natural momentumspace calculation of the Chern number (which can be used here because the honeycomb lattice is periodic). For the fractal lattice (Fig. 3b), the result of the realspace calculation is interesting, as there are many quasienergy values having nonzero realspace Chern numbers. The most important quasienergy range is from −0.05 to 0.05, which is within the topological bandgap of the helical honeycomb lattice where the realspace Chern number is 1. As shown in Fig. 3b, in the helical fractal lattice, the quasienergies in the range between −0.05 and 0.05 correspond to real space Chern number 1, hence supporting the observation that edge states in this range (e.g., state numbers 95 and 93) are indeed topological. We find that the gaps between regions of eigenvalues around −0.5, −0.2, 0.2 and 0.5, shaded in gray in Fig. 2b, separate different bunches of “bulk states” (states residing away from the edges) with a realspace Chern number of 0, which means that these gaps are topologically trivial.
Having found unidirectional edge states with the realspace Chern number 1, we study the evolution of the edge states in evolution simulations in the presence of defects and disorder. Specifically, to verify that edge state number 95 is indeed topological, we demonstrate its ability to display topologically protected transport, the hallmark of topological physics. We launch a wavepacket at the edge of the fractal lattice and simulate its propagation (Fig. 4a–e). The initial wavepacket is a superposition of eigen edge states such that it has a finite width (see Fig. 4a). Figure 4b–e shows the light intensity at different propagation distances Z = 10, 20, 30, 40 cm. Clearly, the wavepacket moves along the edge of the fractal lattice and passes the corner without scattering. During propagation, the wavepacket remains confined to the edge, not penetrating into the bulk and backscattering. Next, we test the robustness to disorder. The simulation in Fig. 4f–j shows that the wavepacket can pass a defect (indicated by the blue dot in the fractal lattice)—a site with onsite disorder of strength \(0.1c_0\). We find that the propagation of wavepackets of edge states in the fractal system is very robust against random onsite disorder of strength up to \(0.2c_0\). The only visible difference between the initial and final wavepackets is the diffraction broadening caused by dispersion (because the edge states comprising the wavepacket evolve at slightly different rates).
The topologically protected transport of edge states in the fractal lattice is not unique to the outer edge. Supplementary Movie #1 shows a similar simulation for an inner edge in the fractal lattice (the perimeter of a hole). The excited edge state number 93 exhibits robust evolution, in the same vein as for the outer edge of the fractal lattice. Since highergeneration fractals always include more inner edges as the generation increases, we find (in simulations) that they exhibit robust propagation on the inner edges—as long as the edge includes an area that is larger than G(3)—to serve as the “bulk” region for the respective inner edge.
Altogether, we have shown that the fractal lattice of helical waveguides has a nondegenerate unidirectional edge state residing in a gap (Fig. 2), that several edge states have a realspace Chern number of 1 (Fig. 3), and that wavepackets made up of these edge states (in both the outer and inner edges) display robust transport by going around the corner and passing defects without backscattering or scattering into the bulk. Hence, we proved that the fractal lattice acts as a topological insulator, although there is no bulk whatsoever, and that one cannot rely on bulkedge correspondence.
At this point, it is very important to emphasize that there are key differences between the fractal lattice and a helical honeycomb lattice with randomly missed sites. As we show in the Supplementary Information, Section B, a honeycomb lattice with randomly missed sites is not a topological insulator: its realspace Chern number is always in the proximity of zero, and its “edge states” do not exhibit unidirectional robust transport. It is clear that the additional symmetries of selfsimilarity on multiple scales, which are at the heart of fractality, are crucial for the existence of topological features in driven fractal lattices.
Finally, we study a hybrid lattice combining both the fractal and honeycomb lattices stitched together, as shown in Fig. 5. We launch a wavepacket comprised of topological edge states on the honeycomb side and simulate its propagation into the fractal side of the lattice. Had our lattice been strictly honeycomb, this wavepacket would propagate without scattering into the bulk and without backscattering even in the presence of disorder (or defects)—as long as the amplitude of the disorder (defect) does not close the topological gap. However, our lattice here is a hybrid: halfhoneycomb, halffractal. Hence, this numerical experiment will serve to show whether (or not) the edge states we have found support topologically protected transfer from honeycomb to fractal lattices modulated by the same helicity. The launched wavepacket shown in Fig. 5a is constructed from a superposition of edge states of the honeycomb lattice. Figure 5b–d shows the evolution, displaying the light intensity distributions at several propagation distances Z = 5, 10, 15 cm. The wavepacket moves along the edge of the honeycomb lattice, passes the corner without scattering, enters the fractal lattice and continues moving along the edge of the fractal lattice. Throughout propagation in the hybrid lattice, the wavepacket remains confined to the edge, does not penetrate into the bulk and does not exhibit backscattering. Moreover, the simulation in Fig. 5e–h shows that the wavepacket is able to pass a defect in the fractal lattice (its position is given by the blue dot)—a site with onsite disorder of strength \(0.1c_0\). Supplementary Movies #2 and 3 show longterm propagation in this hybrid lattice, with the wavepacket encircling the lattice multiple times. Supplementary Movie #4 shows the transport with the wavepacket initially launched at the fractal lattice. Finally, Fig. S3 shows the propagation of a wavepacket in a hybrid lattice where the two components possess different nonzero real space Chern numbers. In this nonmatched semifractal lattice, the wave partially moves along the edge and partially penetrates into the “bulk” of the fractal lattice, which indicates that this system has no topological protection. That is, for a hybrid semifractal system to be topological, its constituents should have the same realspace Chern number.
Discussion
As stated earlier, our findings here are in fact a prelude to upcoming experiments in a photonic platform, which will provide experimental proof that fractal lattices can indeed behave as topological insulators. It is therefore essential to carry out wave dynamics simulations with the actual experimental parameters and examine the evolution. As shown in Supplementary Information Fig. S4, we simulate the wave dynamics of the tightbinding example of Fig. 5. Our wave dynamics simulations show good agreement with the tightbinding simulations, suggesting that what we propose here is readily experimentally accessible with the current technology.
In summary, we proposed photonic Floquet topological insulators in a fractal lattice and demonstrated robust transport along the outer and inner edges of the fractal landscape. We underpinned the difference between driven (helical) fractal lattices and lattices with randomly missed sites and showed that fractal symmetries are crucial for the existence of topological features. Finally, we showed that topological edge states can pass from the edge of the honeycomb lattice to the outer edge of the fractal structure (of the same chirality) virtually without scattering, despite the transition from twodimensions to a fractal dimension. The parameters used in this work are all readily accessible for experiments with photonic lattices fabricated using direct laser writing^{13,23}. These experiments could be the first experimental realization of topological fractal insulators^{30,31,32}.
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Acknowledgements
This work was sponsored by the Israel Science Foundation, by the US Air Force Office of Scientific Research (AFOSR), and by an Advanced Grant from the European Research Council.
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Yang, Z., Lustig, E., Lumer, Y. et al. Photonic Floquet topological insulators in a fractal lattice. Light Sci Appl 9, 128 (2020). https://doi.org/10.1038/s4137702000354z
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DOI: https://doi.org/10.1038/s4137702000354z
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