Abstract
The discovery that the band structure of electronic insulators may be topologically nontrivial has revealed distinct phases of electronic matter with novel properties^{1,2}. Recently, mechanical lattices have been found to have similarly rich structure in their phononic excitations^{3,4}, giving rise to protected unidirectional edge modes^{5,6,7}. In all of these cases, however, as well as in other topological metamaterials^{3,8}, the underlying structure was finely tuned, be it through periodicity, quasiperiodicity or isostaticity. Here we show that amorphous Chern insulators can be readily constructed from arbitrary underlying structures, including hyperuniform, jammed, quasicrystalline and uniformly random point sets. While our findings apply to mechanical and electronic systems alike, we focus on networks of interacting gyroscopes as a model system. Local decorations control the topology of the vibrational spectrum, endowing amorphous structures with protected edge modes—with a chirality of choice. Using a realspace generalization of the Chern number, we investigate the topology of our structures numerically, analytically and experimentally. The robustness of our approach enables the topological design and selfassembly of noncrystalline topological metamaterials on the micro and macro scale.
Main
Condensedmatter science has traditionally focused on systems with underlying spatial order, as many natural systems spontaneously aggregate into crystals. The behaviour of amorphous materials, such as glasses, has remained more challenging^{9}. In particular, our understanding of common concepts such as bandgaps and topological behaviour in amorphous materials is still in its infancy when compared with crystalline counterparts. This is not only a fundamental problem; advances in modern engineering, both of metamaterials and of quantum systems, have opened the door for the creation of materials with arbitrary structure, including amorphous materials. This prompts a search for principles that can apply to a wide range of amorphous systems, from interacting atoms to mechanical metamaterials.
In the exploration of topological insulators, conceptual advances have proven to carry across between disparate physical realizations, from quantum systems^{10}, to photonic waveguides^{11}, to acoustical resonators^{12,13}, to hinged or geared mechanical structures^{3,14}. One promising model system is a class of mechanical insulators consisting of gyroscopes suspended from a plate. Appropriate crystalline arrangements of such gyroscopes break timereversal symmetry, opening topological phononic bandgaps and supporting robust chiral edge modes^{5,6}.
Unlike trivial insulators, whose electronic states can be thought of as a sum of independent local insulating states, topological insulators require the existence of delocalized states in each nontrivial band and prevent a description in terms of a basis of localized Wannier states^{15,16,17}. It is natural, therefore, to assume that some regularity over long distances may be key to topological behaviour, even if topological properties are robust to the addition of disorder. However, the extent to which spatial order needs to be built into the structure that gives rise to topological modes is unclear. We report a recipe for constructing amorphous arrangements of interacting gyroscopes—structurally more akin to a liquid than a solid—that naturally support topological phonon spectra. By simply changing the local connectivity, we can tune the chirality of edge modes to be either clockwise or counterclockwise, or even create both clockwise and counterclockwise edge modes in a single material. This shows that topology, a nonlocal property, can naturally arise in materials for which the only design principle is the local connectivity. Such a design principle lends itself to imperfect manufacturing and selfassembly. Although our construction arises naturally in mechanical metamaterials, we show that it extends to electronic systems in the tightbinding limit.
The first local construction we consider is shown in Fig. 1a. Starting from an arbitrary point set, a natural way to form a network is to generate a Voronoi tessellation, either via the Wigner–Seitz construction or by connecting centroids of a triangulation^{18}. Treating the edges of the cells as bonds and placing gyroscopes at the vertices leads to a network reminiscent of ‘topological disorder’ in electronic systems^{19}. A range of frequencies arises in which all modes are tightly localized, and this frequency region overlaps with the corresponding bandgap of the honeycomb lattice. Crucially, we find that gyroscopeandspring networks constructed in this way from arbitrary initial point sets invariably have such a mobility gap in a frequency range determined by the strength of the gravitational pinning and spring interactions.
Our networks are reminiscent of ‘topologically disordered’ electronic systems^{19}. In these systems, a central characteristic is that the local density of states as a function of frequency is predictive of the global density of states. Specifically, bandgaps or mobility gaps are preserved^{19,20,21}. Interestingly, we find that, even in the presence of band topology, averaging the local density of states over mesoscopic patches (~10 gyroscopes) reproduces the essential features of the global density of states as a function of frequency. Furthermore, we find that inserting mesoscopic patches of our structures into a variety of other dissimilar networks (see Supplementary Figs. 8–10) does not significantly disrupt the averaged local density of states of the patch.
Crucially, we find that our structures show hallmarks of nontrivial topology. When the system is cut to a finite size, modes confined to the edge populate the mobility gap, mixed in with localized states. As shown in the direct simulations of Fig. 1b and Supplementary Videos 1 and 2, shaking a gyroscope on the boundary results in chiral waves that bear all the hallmarks of protected edge states (robustness to disorder and absence of backscattering).
An experimental realization can be readily constructed from gyroscopes interacting magnetically, as seen from below in Fig. 1c. As in ref. ^{5}, these gyroscopes consist of threedimensionalprinted units encasing d.c. motors that interact via magnetic repulsion. Probing the edge of this system immediately generates a chiral wavepacket localized to the boundary, confirming that this class of topological material is physically realizable and robust (Fig. 1c and Supplementary Videos 3 and 4).
This behaviour begs for a topological characterization, even though it might be surprising that topology can emerge from such a local construction. The existence of chiral edge states in an energy gap is guaranteed if an invariant known as the Chern number is nonzero, and the direction of the chiral waves is given by its sign. Although the Chern number was originally defined in momentum space, several generalizations have been constructed in coordinate space in order to accommodate disorder in crystalline electronic materials^{22,23,24}. In these methods, information about the system’s vibrations above a cutoff frequency, ω_{c}, is carried by the projection operator, P. Each element P_{ ij } measures the response of gyroscope j to excitations of gyroscope i within a prescribed range (band) of frequencies.
According to one such method, proposed in ref. ^{22}, a subset of the system is divided into three parts and labelled in a counterclockwise fashion (red, green and blue regions in Fig. 2). These regions are then used to index components of an antisymmetric product of projection operators:
The sum of such elements converges to the Chern number of the band above a chosen cutoff frequency, ω_{c}, when the summation region has enclosed many gyroscopes (see Supplementary Videos 5 and 6 and Supplementary Information).
Equation (1) can be understood as a form of charge polarization in the response of an electronic material to a locally applied magnetic field. Applying a magnetic field to a small region of a material induces an electromotive force winding around the site of application. If the material is a trivial insulator, any changes in charge density there arise from local charge rearrangements, which result in no accumulation of charge. By contrast, a topological electronic system has a Hall conductivity determined by the Chern number. As a result, a net current will flow perpendicular to the electromotive force, inducing a netnonzero charge concentrated at the magnetic field site, compensated by charge on the boundary. As we show in the Supplementary Information, the amount of local charging is proportional to the applied field, and the proportionality constant is the Chern number of equation (1).
Figure 2a shows the results of equation (1) computed for the Voronoized networks. As the cutoff frequency for the projector is varied (here it is lowered from 4Ω_{g}), the computed Chern number converges to ν = −1 when all extended states in the top band lie above the cutoff frequency, confirming that the modes observed in Fig. 1b,c are topological in origin and predicting their direction. The Chern number remains at its value of ν = −1 for a broad range of frequencies in which any existing states are localized, and thus do not contribute to the Chern number. The Chern number returns to zero once more conductancecarrying extended states are included in the calculation.
Having established this connection, we now discuss how the Chern number can be controlled. In particular, we show that by considering alternative decorations of the same initial point set, it is possible to flip the chirality of the edge modes or even provide multiple gaps with differing chirality. One possible construction arises naturally from joining neighbouring points in the original point set, leading to a Delaunay triangulation. As shown in Supplementary Video 7, such networks show no gaps and no topology, suggesting that the local geometry dictated by Voronoization is responsible for its emergent topology. A clue can be found by noting that the Voronoized networks are locally akin to a honeycomb lattice. The honeycomb is the simplest lattice with more than one site per unit cell, a necessary condition for supporting a bandgap in a lattice. Moreover, this lattice was previously found to be topological with the same Chern number^{5}.
Building on this insight, we introduce a second decoration, which we dub ‘kagomization,’ shown in Fig. 3a. If applied to a triangular point set, Voronoization produces a honeycomb lattice and kagomization produces a kagome lattice, the simplest lattice with three sites per unit cell, which we have found to produce ν = +1 gyroscopic metamaterials. Proceeding as with the Voronoized network case (Fig. 3b,c), we find the presence of topologically protected modes with opposite direction and the corresponding opposite Chern number in the the band structure (see Supplementary Video 8). Other local constructions, such as the ‘spindle’ networks in Fig. 3d,e provide multiple mobility gaps, each with a different edge mode chirality, offering a transmission direction tuned by frequency (see Supplementary Video 9).
One might think there could be a mapping from the geometry of each vertex to the chirality of the edge modes. However, taken together, our Voronoized, kagomized and spindle networks demonstrate that simply counting nearest neighbours is not sufficient to determine the topology: a description beyond nearest neighbours is required (see also Supplementary Fig. 29). On the other hand, we are able to change the Chern number of a structure via local decorations. To uncover the extent to which a network’s topology is stored locally, consider the projection operator, P. The projector value P_{ ij } measures the vibrational correlation between gyroscope j and gyroscope i when considering all modes above a cutoff frequency. By explicitly computing its magnitude in our networks, we find that the magnitude of P_{ ij } falls off exponentially with distance (Supplementary Fig. 11). Remarkably, explicitly cutting out a section of the network and embedding it in a network with a different spectrum results in only a slight change to the local projector values (<2%) (see Supplementary Fig. 12). Since the Chern number is built from these projector elements, it then follows that the local structure of the gyroscope network, combined with some homogeneity of this local structure across the lattice, is all that is needed to determine the Chern number (see Supplementary Video 10 and the Supplementary Information section entitled ‘Sign of Chern number from network geometry’).
This situation is reminiscent of electronic glasses, in which the local binding structure gives rise to a local ‘gap.’ Under weak assumptions of homogeneity, this gap can be shown to extend to the whole system^{19,20}. The case with topology is similar: the nextnearestneighbour angles in a network’s cell open a local ‘gap’ by breaking timereversal symmetry.
For amorphous networks, we make the correspondence between the bulk topological invariant and the edge states on the boundaries by considering a gyroscopic sample shaped into an annulus (see refs ^{25,26,27}). Adiabatically tuning the interactions between pairs of gyroscopes along a radial cut (by adding a fixture to one gyroscope from each pair) pumps each edge mode into a neighbouring mode, as shown in Supplementary Video 11 and in the Supplementary Information section ‘Spectral flow through adiabatic pumping’. If we consider all states below a gap cutoff frequency, the process—which mimics the effect of threading a magnetic field through the centre of an annulus in an electronic system—trades one state localized on the outer boundary for an extra state on the inner boundary of the annulus, which we connect to the realspace Chern number (equation (1)) in the Supplementary Information.
As in the lattice case, a mobility gap becomes topological due to timereversal symmetry breaking: bond angles in these networks are not multiples of π/2 (ref. ^{5}). We can probe this mechanism by eliminating a gap’s topology. Alternating the gravitational precession frequency, Ω_{g}, of neighbouring gyroscopes in a network mimics the breaking of inversion symmetry on a local scale, an effect that competes against the timereversal gap opening mechanism. When the precession frequency difference between sites is large enough, this competing mechanism eliminates edge modes, triggering a transition to a ν = 0 mobility gap, shown in Fig. 4a and Supplementary Videos 12 and 13.
Equipped with these insights, we can easily engineer networks that are heterogeneous mixtures of multiple local configurations. Figure 4b–d highlight some results of combining Voronoized and kagomized networks or encapsulating one within another. As the Voronoized and kagomized networks share a mobility gap, excitations are localized to their interface, offering a method of creating robust unidirectional waveguides, such as the sinuous waveguide shown in Fig. 4b and Supplementary Video 14. Figure 4c demonstrates that additional topological mobility gaps at higher frequency in the kagomized network allow bulk excitations to be confined to an encapsulated Voronoized region (see also Supplementary Video 15). Random mixtures of the two decorations, shown in Fig. 4d, demonstrate heterogeneous local Chern number measurements (red for ν=+1 and blue for ν=–1), with mobilitygap excitations biased toward the interfaces between red and blue regions (see Supplementary Information and Supplementary Video 16).
As our networks are structurally akin to liquids, they support topological modes in the absence of longrange spatial order. The details of the underlying point set are not essential, and neither are the details of the local Voronoization or kagomization procedures. We verified this by replacing the centroidal construction^{18} with a Wigner–Seitz construction (see Supplementary Information Figs. 24–26 for a comparison). Beyond mechanical materials, we find similar results in electronic tightbinding models of amorphous networks, underscoring the generality of the finding (Supplementary Fig. 32).
This study demonstrates that local interactions and local geometric arrangements are sufficient to generate chiral edge modes, promising new avenues for engineering topological mechanical metamaterials generated via imperfect selfassembly processes. Such selfassembled materials could be constructed, for instance, with micrometrescale spinning magnetic particles. Since our methods bear substantial resemblance to tightbinding models, our results also find direct application not only to electronic materials, as we have demonstrated, but also to photonic topological insulators^{11}, acoustic resonators^{12,13} and coupled circuits^{28}.
Note added in proof: After the submission of this work, we became aware of a concurrent theoretical study of the existence of amorphous electronic topological insulators in two and three dimensions^{29}.
Methods
For small displacements (δX, δY) of the pivot points from vertical alignment with the centres of mass, the equation of motion for a gyroscope takes the form
where \({\Omega }_{{\rm{k}}}=k{\ell }^{2}{\rm{/}}(I\omega )\), and \({\Omega }_{{\rm{g}}}=\ell mg{\rm{/}}(I\omega )\), while k is the spring constant, \(\ell \) is the length of the pendulum from the pivot to the centre of mass, I is the moment of inertia along the spinning axis, ω is the angular spinning frequency, m is the mass of the gyroscope and g is the gravitational acceleration. The simulations evolve equation (2) using a Runge–Kutta fourthorder explicit method run on a graphics processing unit using OpenCL.
To obtain spectra and normal modes, note that equation (2) defines the entries for a system’s dynamical matrix, D, such that
where the components of \({\boldsymbol{\psi }}\) contain information on the displacements of the gyroscopes. To map to a tightbinding model, it is useful to write the displacement of the ith gyroscope as ψ_{ i } ≡ δX_{ i } + iδY_{ i } and note that the eigenvalue problem gives pairs of eigenvalues ±ω, so that
Then
The experiment in Fig. 2c shows a chiral wavepacket localized to the boundary in an experimental realization of 195 gyroscopes in a Voronoized network based on a point set generated from a jammed packing. Each gyroscope consists of a spinning motor (~150 Hz) housed in a threedimensionalprinted enclosure (as in ref. ^{5}), and each gyroscope is suspended from an acrylic plate by a spring, an attachment method that was found to reduce damping.
To establish the equivalence of equation (1) and the response to a pointlike magnetic field, we study the effect of a perturbation on the projection operator \({P}_{ij}\equiv {\sum }_{n\in {\rm{band}}}{\chi }_{n}(i){\chi }_{n}^{* }(j),\) where χ are the perturbed wavefunctions. In the Supplementary Information, we link the change in the diagonal elements to the charge accumulation near an applied magnetic field:
where Δρ is the change in charge density where the magnetic field is applied, B_{ z } is the magnetic field normal to the sample and ν is the Chern number of the occupied bands when the sample is periodic.
Having shown that the Chern sum (equation (1)) is equal to the charge accumulated when a quantum of magnetic flux is inserted, we can establish a correspondence between the bulk invariant and edge modes on the boundary by introducing a hole at the site of insertion. The realspace Chern number is then equal to the number of edge states that accumulate on the inner boundary as an effective magnetic flux is introduced through the hole^{25,26,27}. The effective magnetic flux is manifest as a phase shift in the interactions for any loop of spring connections that encloses the hole. We construct this phase shift by altering the subset of the nearestneighbour interactions that traverse a cut of the annulus, such that the force of one gyroscope on its neighbour across the cut is altered by a rotation
In the Supplementary Information, we propose a concrete picture of how this could be built in an experiment by attaching an extensible ring to a small number of gyroscopes.
To see topological robustness in a simpler situation, we find similar behaviour in an amorphous electronic tightbinding model using the model Hamiltonian
where \(\left\langle ij\right\rangle \) denotes nearest neighbours ij and \(\left\langle \left\langle ij\right\rangle \right\rangle \) denotes pairs of nextnearest neighbours (NNN). The parameter t_{2} tunes the strength of all NNN hoppings, and ϕ_{ ij } controls the degree to which the hopping i → j breaks timereversal symmetry (by tuning the imaginary term). As shown in the Supplementary Information, topological edge modes arise in amorphous tightbinding lattices, whether the NNN hopping is uniform or bond angledependent.
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Acknowledgements
We thank M. Levin, C. Kane and E. Prodan for useful discussions. This work was primarily supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by National Science Foundation under award number DMR1420709. Additional support was provided by the Packard Foundation. The Chicago MRSEC (US NSF grant DMR 1420709) is also gratefully acknowledged for access to its shared experimental facilities. This work was also supported by NSF EFRI NewLAW grant 1741685.
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James Franck Institute and Department of Physics, The University of Chicago, Chicago, IL, USA
 Noah P. Mitchell
 , Lisa M. Nash
 , Daniel Hexner
 & William T. M. Irvine
Department of Physics, Israel Institute of Technology, Haifa, Israel
 Ari M. Turner
Enrico Fermi Institute, The University of Chicago, Chicago, IL, USA
 William T. M. Irvine
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Contributions
W.T.M.I. and N.P.M. designed research. W.T.M.I. and A.M.T. supervised research. N.P.M. and L.M.N. performed the simulations and experiments. A.M.T., N.P.M., D.H. and W.T.M.I. performed analytical calculations. D.H. contributed numerical tools. N.P.M. and W.T.M.I. wrote the manuscript. All authors interpreted the results and reviewed the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Noah P. Mitchell or William T. M. Irvine.
Supplementary information
Supplementary Video 1
An amorphous gyroscopic network constructed by Voronoizing a point set of a jammed packing exhibits a topological mobility gap. The density of states is shown for a periodic system, with modes coloured by their measured inverse localization length, while the time domain simulation is performed on the same system with open boundary conditions.
Supplementary Video 2
A Voronoized network constructed from a quasicrystalline arrangement of points, here from a rhombic Penrose tiling, likewise exhibits a topological mobility gap. The density of states is shown for a periodic approximant, with modes coloured by their measured inverse localization length, while the time domain simulation is performed on a finite, truly quasicrystalline system.
Supplementary Video 3
In an experimental realization of an amorphous gyroscopic network, a wave packet excited at the edge with a frequency centred in the mobility gap propagates around the entire system. The colours of the gyroscopes’ excitations denote the phase, and the size of the coloured circles are proportional to their displacement.
Supplementary Video 4
In a larger experimental realization of an amorphous gyroscopic network, a wave packet excited at the edge with a frequency centred at the mobility gap travels clockwise around the boundary of the system, regardless of the where the excitation is initiated on the boundary. The colours of the gyroscopes’ excitations denote the phase, and the size of the coloured circles are proportional to their displacement.
Supplementary Video 5
Visualization of a realspace Chern number calculation on a honeycomb lattice demonstrates convergence to within ~15% of the target value (ν = −1 for the lattice’s upper band) after the summation region has a radius of approximately three lattice spacings.
Supplementary Video 6
Realspace measurement of the Chern number for a Voronoized amorphous network converges in a similar fashion to the lattice case. The network shown here is constructed from a hyperuniform point set.
Supplementary Video 7
A triangulated amorphous network of gyroscopes does not exhibit chiral edge modes. This highlights that placing gyroscopes at the nodes of an arbitrary spring network does not generally give rise to topological behaviour. The density of states is shown for a periodic system, with modes coloured by their measured inverse localization length, while the time domain simulation is performed on the same system with open boundary conditions.
Supplementary Video 8
An amorphous gyroscopic network constructed by kagomizing a point set of a jammed packing also exhibits topological mobility gaps. The density of states is shown for a periodic system, with modes coloured by their measured inverse localization length, while the time domain simulation is performed on a system with open boundary conditions.
Supplementary Video 9
A spindle network, constructed using the decoration in Fig. 3d of the main text, exhibits both clockwise and anticlockwise edge modes. The two topological gaps are separated in frequency, enabling transmission with frequencydependent chirality using a single material.
Supplementary Video 10
Nesting a patch of a honeycomb lattice within a kagome lattice results in a measurement of the Chern number that flips sign when gyroscopes beyond the interface are included.
Supplementary Video 11
To demonstrate spectral flow in an amorphous gyroscopic metamaterial, modify the spring attachments along a cut of an annular sample (blue dashed line). The springs are attached to an extensible ring on the gyroscopes immediately above the cut. The location of the spring attachment is given by the gyroscope’s current displacement rotated by a phase. The gap modes in the spectrum localized to the inner boundary of the annulus rise in frequency as phase increases, while the modes on the outer edge decrease. Once the attachment point ‘leads’ the gyro’s displacement by a full rotation, the spectrum returns to its original form, but each edge state has been pumped into an adjacent state.
Supplementary Video 12
By increasing the difference in gravitational precession frequencies between neighbouring sites (coloured white and black for increased and decreased frequencies), the mobility gap is closed, then reopened. When reopened, the Chern number difference between bands is zero.
Supplementary Video 13
After detuning the gravitational precession frequencies of neighbouring gyroscopes (coloured white and black for increased and decreased frequencies, respectively), there are no chiral edge modes in the system. A single gyroscope on the lower edge of the sample is shaken at a frequency in the middle of the spectrum, but the excitation has no chirality and is not confined to the edge of the sample.
Supplementary Video 14
Combining kagomized (upper) and Voronoized (lower) networks at an arbitrary boundary — here taken to be a boundary spelling ‘CHERN’ — provides a unidirectional waveguide. Shaking a single gyroscope at the left sends an excitation confined to the sinuous boundary across the sample.
Supplementary Video 15
Additional topological mobility gaps at higher frequencies in the kagomized network allow bulk excitations to be confined to an encapsulated Voronoized region. The density of states is coloured by participation ratio.
Supplementary Video 16
Random mixtures of the two decorations demonstrate heterogeneous, spatially resolvable Chern number measurements.
Supplementary Information
Supplementary Information, Supplementary Figs 1–32, Notes on Supplementary Videos, Supplementary References
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