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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 ${\boldsymbol{A}}$.
Fig. 1 Iterative generations of the Sierpinski gasket (SG).
The first generation G(1) of the SG has nine blue sites. G(n) has three copies of G(n−1), sharing three corner sites. The fractal lattice of helical waveguides is generation G(4) with a total of 204 sites. Each blue site marks the position of a helical waveguide. The helicity of the waveguides introduces an artificial vector potential . For presentation simplicity, we draw only the threedimensional schematic of a G(1) lattice of helical waveguides${\boldsymbol{A}}\left(z \right) = A_0\left[ {\sin \left({{\mathrm{\Omega }}z} \right),  \cos \left({{\mathrm{\Omega }}z} \right), 0} \right]$ The equation governing the diffraction of light in this fractal photonic lattice under the tightbinding approximation^{13} can be written as
$$ i\partial _z\psi _n = c_0\mathop {\sum }\limits_{<{m}>} e^{i{\boldsymbol{A}}\left( z \right) \cdot {\boldsymbol{r}}_{m,n}}\psi _m $$ (1) 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.
Fig. 2 Eigenstates of the fractal lattice.
a Energy spectrum with (straight waveguides). The spectrum of this nontopological system displays a large central gap (large gray region), with a flat band in the midgap made up of immobile degenerate states. b Quasienergy spectrum with${\boldsymbol{A}}\left(z \right) = 0$ and${\boldsymbol{A}}\left(z \right) \ne 0$ . The inset shows an enlarged view of the center box. The shaded regions mark quasigaps: regions within which there are no eigenstates. In this topological fractal system, the edge states from either side of the flat band evolve into nondegenerate unidirectional edge states. c Field intensity patterns of two eigenstates localized at the external and internal edges of the fractal lattice (states 93 and 95, gray and red dots, respective√ly). The color bar indicates the intensity (normalized to the peak intensity in each state). The parameters used are the ambient refractive index$A_0 = kR{\mathrm{\Omega }}$ , coupling strength$n_0 = 1.45$ , wavelength$c_0 = 1.9\, {\rm{cm}}^{  1}$ µm, helix radius$\lambda = 0.633$ µm, longitudinal frequency of the helix$R = 10$ , and lattice constant${\mathrm{\Omega }} = 2\pi \, {\rm{cm}}^{  1}$ µm$a = 14\sqrt 3$ 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
$$ C = 12\pi i\mathop {\sum }\limits_{j \in A} \mathop {\sum }\limits_{k \in B} \mathop {\sum }\limits_{l \in C} \left( {P_{jk}P_{kl}P_{lj}  P_{jl}P_{lk}P_{kj}} \right) $$ (2) 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.
Fig. 3 The realspace Chern number as a function of the quasienergy for the honeycomb (a) and the fractal (b) lattices with the same nonzero artificial gauge field.
The lower panels present the honeycomb and the fractal lattices, which are both triangularshaped. When calculating the realspace Chern number, the hexagons are divided into three regions with different shades of gray, each enclosing many waveguides. The center inset shows an enlarged view of the quasienergy range of the fractal lattice from −0.1 to 0.1. The parameters of the lattices are the same as in Fig. 2Having 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).
Fig. 4 Tightbinding simulations of an edge wavepacket propagating in a fractal lattice.
a–e Evolution of topological edge states in the fractal SG(4) lattice. a Intensity distribution of the initial field constructed from a truncated topological edge state in the fractal lattice. b–e Intensity distribution at propagation distances Z = 0, 10, 20, 30, 40cm. f–j Evolution in the fractal lattice containing an onsite disorder of , the position of which is marked by the blue dot. The wavepacket displays topologically protected edge transport around the corners and is unaffected by the disorder. The color bar indicates the field intensity. The parameters for the numerical simulation are the same as in Fig. 2$0.1c_0$ 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.
Fig. 5 Tightbinding simulations of an edge wavepacket propagating in a hybrid lattice consisting of fractal and honeycomb lattices.
The initial wavepacket (a, e) is constructed from a truncated edge state of the honeycomb lattice. a–d Propagation from the honeycomb region into the fractal region, displayed at propagation distances Z=0, 5, 10, 15cm. f–h Propagation into the fractal region containing a defect (blue dot) in the form of onsite disorder of . The wavepacket exhibits propagation along the edges and around the corners and bypassing disorder. The color bar indicates the field intensity. The parameters for the numerical simulations are the same as in Fig. 2$0.1c_0$