Amorphous photonic topological insulator

Photonic topological insulators (PTIs) exhibit robust photonic edge states protected by band topology, similar to electronic edge states in topological band insulators. Standard band theory does not apply to amorphous phases of matter, which are formed by non-crystalline lattices with no long-range positional order but only short-range order. Among other interesting properties, amorphous media exhibit transitions between glassy and liquid phases, accompanied by dramatic changes in short-range order. Here, we experimentally investigate amorphous variants of a Chern-number-based PTI. By tuning the disorder strength in the lattice, we demonstrate that photonic topological edge states can persist into the amorphous regime, prior to the glass-to-liquid transition. After the transition to a liquid-like lattice configuration, the signatures of topological edge states disappear. This interplay between topology and short-range order in amorphous lattices paves the way for new classes of non-crystalline topological photonic materials.

experimentally extended a Chern-number-based PTI 10-13 into the amorphous regime. Similar to previous theoretical proposals 33, 34 , the amorphous PTI that we study consists of gyromagnetic rods that are arranged in computer-generated amorphous lattice patterns, and magnetically biased to break time-reversal symmetry. By performing edge/bulk transmission and near-field distribution measurements, we experimentally verify the existence of robust topological edge states in the amorphous PTIs prior to the onset of the glass transition. When the lattice undergoes the glass transition, the local site connectivity is dramatically altered, resulting in the closing of the bulk topological gap and the disappearance of the topological edge states. Although the concept of amorphous topological insulators has been theoretically proposed in condensed matter systems 35,36 , and some related features have been realized in a mechanical network of gyroscopic oscillators 37 , there has never been any systematic experimental study of how band topological effects depend on short-range order (including the important role of the glass transition). This work thus enriches our understanding of topological photonic materials, and paves the way to exploring new types of photonic lattices that can host topologically protected edge states.

Results
Photonic lattices with different structural correlations are generated using numerical particle-packing methods previously developed in soft condensed matter physics [38][39][40] . The packing is conducted in a two-dimensional (2D) square unit cell with periodic boundary conditions, bidisperse discs (radius ratio 1.2 with equal distributions; see Supplementary Information). The process ends upon reaching a target packing density ϕ (the fraction of space covered by the discs).
The configuration is then transformed into a photonic lattice by replacing the discs with gyromagnetic cylindrical rods (see Figs.1a,b; note that the crystalline lattice is built on a triangular lattice not generated by the packing method). We define a disorder index (DI) as DI = (ϕ max -ϕ)/ ϕ max , where ϕ max = 0.9069 is the densest possible packing density (corresponding to a triangular lattice). As defined, the DI is positively related to the amount of disorder in the lattice.
where r is the distance between a pair of gyromagnetic rods, N is the number of rods, and N L a  , where L is the size of the lattice system. This quantifies the degree of structural correlation in the lattice and has been extensively employed to characterize amorphous phases 41 .
The pair correlation functions for different lattice structures are plotted in Fig. 1c. For DI = 0, g(r) shows sharp peaks in the whole r range. For DI = 0.1, the first peaks in g(r) split into subpeaks due to the bidisperse packing; the shoulders of the second peak indicate local clustering, a common phenomenon in amorphous materials 42 ; the other peaks progressively damp away at r/a > 3, indicating the lack of long-range order. For DI = 0.45, the short-range order decreases, and the first g(r) peak is less than half of the counterpart in the DI=0.1 case. For the weakly-correlated lattice with DI = 0.8, there is only one visible peak and g(r) ~ 1 over most of the range, indicating weak short-range order (i.e., a liquid-like lattice configuration).
These variations in lattice properties can have significant impacts on band topological phenomena. In the following content, we will start with the topological bandgap of a crystalline PTI corresponding to a DI=0 lattice (left panel in Fig. 1d). We will demonstrate that this topological gap persists for amorphous lattices up to DI= 0.45 (middle panels in Fig. 1d), before the glass transition. After the glass transition, the topological frequency gap closes (right panel in  Next, we fabricate an amorphous PTI with DI = 0.1 and characterize it using the same experimental setup. In the bulk transmission measurements, we observe a significant dip in both forward and backward transmission between 10.6 GHz to 11.4 GHz, indicating a mobility gap at frequencies close to the crystalline counterpart (Fig. 2d). In the edge measurements, we observe a huge difference between forward and backward transmission in the frequency range of the mobility gap (Fig. 2d). Mapping out the field distributions reveals a unidirectional edge state propagating clockwise. These experimental results are consistent with numerical calculations (see Fig. 1d and Supplementary Information), indicating that the localization length is extremely short from ~10.5 GHz to ~11.5 GHz. Since the amorphous PTI lacks periodicity, it lacks a properly defined momentum-space bandstructure; to characterize the topology, we adapt the Bott index (see Supplementary Information), which acts like the Chern number but can be applied in real space 27,29 . As shown in Fig. 2a, the Bott index has a nontrivial value of 1 (equivalent to the Chern number for the earlier crystalline PTI) within the mobility gap. All of these results -the bulk gap, one-way chiral edge transport, first-principles calculations of the mobility gap, and the Bott index -are in excellent agreement, pointing to the existence of topologically protected edge states in the amorphous PTI. To verify the robustness of the edge states in the amorphous PTI, we introduce defects along the edges. Two types of defects were tested. In the first case, a rectangle aluminum obstacle is placed at the edge to block the edge propagation (Fig 3a). In the second case, three gyromagnetic rods are removed to create a large air cavity (Fig. 3b). We then measure the edge transmission. In both cases, we find large differences between forward and backward transmissions in the frequency range of the mobility gap, indicating that the defects do not cause backscattering. Experimental setup is the same as that in Fig. 2. The orange region in (d) denotes the corresponding numerically-calculated topological region.
Next, we study the effects of the glass transition. It should be noted that the nature of the glass transition in real amorphous materials remains poorly understood, despite the extensive theoretical and experimental studies 31, 32 . Using the lattice generation procedure detailed above, As shown in Fig. 4b, CN(r) changes from a step-like curve to a smooth one with increasing DI.
Integrating to the first minimum (a discontinuity) of g(r) gives the coordination number of the nearest neighbors, denoted as CN 1 (inset of Fig. 4b), which represents the average local connectivity of each lattice site. As DI increases, CN 1 drops from a ~ 6 (similar to the crystalline case) to ~2 (similar to a liquid). Around the critical value of DI = 0.45, CN 1 drops very quickly, suggesting a glass transition 31, 44 . Thereafter, CN 1 converges to ~ 2, indicating the completion of the glass transition.
We used first-principle simulations to investigate the interplay between short-range order and topological protection. In the simulations, the photonic lattices are surrounded by PEC boundaries, and a point source is placed near the boundaries. Based on the numerical field distributions, we calculate an empirical parameter 26 where ε is the electromagnetic (EM) energy density, Π is the whole area of the photonic lattice, and Π s is the area one free-space wavelength away from the PEC boundary. When the system hosts topological edge states, they tend to be localized in Π s , so C s is close to unity. The plot of C s versus DI is shown in Fig. 4c. For DI = 0.1, C s is close to unity within the frequency window corresponding to the mobility gap, consistent with the previous results. Upon increasing DI beyond the critical value of 0.45, i.e., around the glass transition, the short-range order quickly decreases, and the frequency window (the high C s region marked in red in Fig. 4c) shrinks rapidly to zero.
To verify these findings experimentally, we fabricate two samples with DI = 0.45 and DI = 0.8, and measure the edge transmission and the electric field distribution using the same setup as in Fig. 2. For DI = 0.45, the topological frequency window shrinks to a narrow range of 11.7 GHz -12.2 GHz (Fig. 4c), and the edge states are only weakly confined to the edge (Fig. 4d). For DI = 0.8, past the glass transition, there is no sign of the topological edge states in the transmission or field distribution measurements; the numerically-calculated localization length shows small fluctuations (Fig. 1d), suggesting the closing of the mobility gap.
We thus experimentally realized amorphous PTIs that lack long-range order but preserve short-range order. Using microwave measurements, we directly observed the bulk mobility gap and the unidirectional propagation of topological edge states, which is robust against defects and disorders. By gradually deforming the amorphous lattice into a liquid-like lattice through the glass transition, we observed the closing of the mobility gap and the disappearance of the topological edge states. These results illustrate the key role of short-range order in the formation of the topological edge states. These insights may be useful for realizing amorphous topological insulators in other physical settings such as acoustics. It would also be interesting to explore other types of non-crystalline photonic topological materials, such as topological random lasers.

Methods
Sample and experimental measurement. The yttrium iron garnet (YIG) ferrite cylinder rods have relative permittivity 13, dielectric loss tangent 0.0002, radius 2.2 mm, and height 4 mm. The saturation magnetization was measured to be M s = 1780 Gauss, and the gyromagnetic resonance loss width to be 35 Oe. In the microwave measurements, a static magnetic field generated by an electromagnet is applied perpendicular to the waveguide, producing a strong gyromagnetic response in the ferrite rods. The spatial non-uniformity of the magnetic field is less than 2% in the sample region. which is the experimentally-obtained value at 11 GHz with a 0.2 T static magnetic field along the -z direction. The effects of dispersion are negligible since both the permittivity and the permeability vary only slightly in the considered frequency band.