Amorphous topological insulators constructed from random point sets

Noah P. Mitchell,1, ∗ Lisa M. Nash,1 Daniel Hexner,1 Ari M. Turner,2 and William T. M. Irvine1, 3, †

1James Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA2Department of Physics, Israel Institute of Technology

3Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA

The discovery that the band structure of elec-tronic insulators may be topologically non-trivialhas revealed distinct phases of electronic matterwith novel properties [1, 2]. Recently, mechan-ical lattices have been found to have similarlyrich structure in their phononic excitations [3,4], giving rise to protected uni-directional edgemodes [5–7]. In all these cases, however, aswell as in other topological metamaterials [3, 8],the underlying structure was finely tuned, be itthrough periodicity, quasi-periodicity or isostatic-ity. Here we show that amorphous Chern in-sulators can be readily constructed from arbi-trary underlying structures, including hyperuni-form, jammed, quasi-crystalline, and uniformlyrandom point sets. While our findings apply tomechanical and electronic systems alike, we focuson networks of interacting gyroscopes as a modelsystem. Local decorations control the topology ofthe vibrational spectrum, endowing amorphousstructures with protected edge modes—with achirality of choice. Using a real-space general-ization of the Chern number, we investigate thetopology of our structures numerically, analyti-cally and experimentally. The robustness of ourapproach enables the topological design and self-assembly of non-crystalline topological metama-terials on the micro and macro scale.

Condensed matter science has traditionally focused onsystems with underlying spatial order, as many natu-ral systems spontaneously aggregate into crystals. Thebehavior of amorphous materials, such as glasses, hasremained more challenging [9]. In particular, our un-derstanding of common concepts such as bandgaps andtopological behavior in amorphous materials is still in itsinfancy when compared to crystalline counterparts. Thisis not only a fundamental problem; advances in modernengineering, both of metamaterials and of quantum sys-tems, has opened the door for the creation of materialswith arbitrary structure, including amorphous materials.This prompts a search for principles that can apply to awide range of amorphous systems, from interacting atomsto mechanical metamaterials.

In the exploration of topological insulators, concep-tual advances have proven to carry across between dis-parate 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 mechanicalinsulators consisting of gyroscopes suspended from aplate. Appropriate crystalline arrangements of such gy-roscopes break time-reversal symmetry, opening topolog-ical phononic band gaps and supporting robust chiraledge modes [5, 6].

Unlike trivial insulators, whose electronic states canbe thought of as a sum of independent local insulat-ing states, topological insulators require the existence ofdelocalized states in each nontrivial band and preventa description in terms of a basis of localized Wannierstates [15–17]. It is natural, therefore, to assume thatsome regularity over long distances may be key to topo-logical behavior, even if topological properties are robustto the addition of disorder. However, the extent to whichspatial order needs to be built into the structure thatgives rise to topological modes is unclear. We report arecipe for constructing amorphous arrangements of in-teracting gyroscopes—structurally more akin to a liquidthan a solid—that naturally support topological phononspectra. By simply changing the local connectivity, wecan tune the chirality of edge modes to be either clock-wise or counter-clockwise, or even create both clockwiseand counter-clockwise edge modes in a single material.This shows that topology, a nonlocal property, can natu-rally arise in materials for which the only design principleis the local connectivity. Such a design principle lendsitself to imperfect manufacturing and self-assembly. Al-though our construction arises naturally in mechanicalmetamaterials, we show that it extends to electronic sys-tems in the tight binding limit.

The first local construction we consider is shown inFigure 1a. Starting from an arbitrary point set, a natu-ral way to form a network is to generate a Voronoi tes-sellation, either via the Wigner-Seitz construction or byconnecting centroids of a triangulation [18]. Treating theedges of the cells as bonds and placing gyroscopes atthe vertices leads to a network reminiscent of ‘topolog-ical disorder’ in electronic systems [19]. A range of fre-quencies arises in which all modes are tightly localized,and this frequency region overlaps with the correspond-ing band gap of the honeycomb lattice. Crucially, wefind that gyroscope-and-spring networks constructed inthis way from arbitrary initial point sets invariably havesuch a mobility gap in a frequency range determined bythe strength of the gravitational pinning and spring in-teractions.

Our networks are reminiscent of ‘topologically disor-

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FIG. 1. Local structure gives rise to chiral edge modes | a Voronoization of an amorphous structure, constructedby connecting adjacent centroids of a triangulation, preserves isotropy and lack of long-range order, here with a hyperuniformpoint set. Two-point correlation functions g(x) (below) reveal isotropic spatial structure for a system of N ≈ 3000 particles.Spatial coordinates x and y are measured in units of the median bond length. b Simulations reveal chiral edge modes in

topological gyroscopic networks. The localization of modes is probed by participation ratio, p =(∑

i |ψi|2)2/N∑

i |ψi|4, andthe density of states is plotted as a function of normal mode oscillation frequency, in units of the gravitational precessionfrequency, Ωg = `mg/Iω. The blue curve overlaying the density of states denotes the frequency of the driving excitation in thesimulation. Here, the characteristic spring frequency, Ωk = k`2/Iω is chosen such that Ωg = Ωk. The inset on the right showsthe amplitude, |δψ|, of the displacement for the single gyroscope which is shaken at a constant frequency. c, An edge modepropagates clockwise in an amorphous experimental gyroscopic network. The motor- driven gyroscopes couple via a magneticdipole-dipole interaction. Despite the nonlinear interaction and spinning speed disorder (∼ 10%), the edge mode appears, nomatter where on the boundary the excitation is initialized.

dered’ electronic systems [19]. In these systems, a cen-tral characteristic is that the local density of states asa function of frequency is predictive of the global den-sity of states. Specifically, band gaps or mobility gapsare preserved [19–21]. Interestingly, we find that, even inthe presence of band topology, averaging the local den-sity of states over mesoscopic patches (∼ 10 gyroscopes)

reproduces the essential features of the global density ofstates as a function of frequency. Furthermore, we findthat inserting mesoscopic patches of our structures into avariety of other dissimilar networks (see SupplementaryInformation Figures 8-10) does not significantly disruptthe averaged local density of states of the patch.

Crucially, we find that our structures show hallmarks

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FIG. 2. Chern number calculations confirm topologi-cal mobility gaps | a, The Chern number is computed forthe band of frequencies above a cutoff frequency, ωc, usinga real-space method. Once all modes in a band that carryHall conductance are included, the Chern number convergesto an integer value. On the left is an overlaid density of statesD(ω) histogram for ten realizations of Voronoized hyperuni-form point sets (∼ 2000 particles), with each mode colored byits inverse localization length, λ−1. The topological mobilitygaps remain in place and populated by highly localized statesfor all realizations. (b and c), The computed Chern numberconverges once ∼20-40 gyroscopes are included in the summa-tion region (red, green, blue regions panel b), and remains atan integer value until the summation region begins to enclosethe sample boundary. All networks have their precession andspring frequencies set to be equal (Ωg = Ωk).

of non-trivial topology. When the system is cut to a finitesize, modes confined to the edge populate the mobilitygap, mixed in with localized states. As shown in thedirect simulations of Figure 1 and Supplementary Videos1 and 2, shaking a gyroscope on the boundary resultsin chiral waves that bear all the hallmarks of protectededge states (robustness to disorder and absence of back-scattering).

An experimental realization can be readily constructedfrom gyroscopes interacting magnetically, as seen frombelow in Figure 1c. Like in [5], these gyroscopes consistof 3D-printed units encasing DC motors which interactvia magnetic repulsion. Probing the edge of this systemimmediately generates a chiral wave packet localized tothe boundary, confirming that this class of topologicalmaterial is physically realizable and robust (Figure 1cand Supplementary Videos 3 and 4).

This behavior begs for a topological characterization,even though it might be surprising that topology canemerge from such a local construction. The existence ofchiral edge states in an energy gap is guaranteed if an in-variant known as the Chern number is nonzero, and thedirection of the chiral waves is given by its sign. Althoughthe Chern number was originally defined in momentum

space, several generalizations have been constructed incoordinate space in order to accommodate disorder incrystalline electronic materials [22–24]. In these meth-ods, information about the system’s vibrations above acutoff frequency, ωc, is carried by the projection operator,P . Each element Pij measures the response of gyroscopej to excitations of gyroscope i within a prescribed range(band) of frequencies.

According to one such method, proposed in [22], a sub-set of the system is divided into three parts and labeledin a counterclockwise fashion (red, green, and blue re-gions in Figure 2). These regions are then used to indexcomponents of an antisymmetric product of projectionoperators:

ν(P ) = 12πi∑j∈A

∑k∈B

∑l∈C

(PjkPklPlj − PjlPlkPkj) . (1)

The sum of such elements converges to the Chern num-ber of the band above a chosen cutoff frequency, ωc, whenthe summation region has enclosed many gyroscopes (seeSupplementary Videos 5 and 6 and Supplementary Infor-mation).

Equation 1 can be understood as a form of charge po-larization in the response of an electronic material to alocally applied magnetic field. Applying a magnetic fieldto a small region of a material induces an electromotiveforce winding around the site of application. If the ma-terial is a trivial insulator, any changes in charge densitythere arise from local charge re-arrangements, which re-sult in no accumulation of charge. By contrast, a topolog-ical electronic system has a Hall conductivity determinedby the Chern number. As a result, a net current will flowperpendicular to the electromotive force, inducing a net-nonzero charge concentrated at the magnetic field site,compensated by charge on the boundary. As we showin the Supplementary Information, the amount of localcharging is proportional to the applied field, and the pro-portionality constant is the Chern number of Equation 1.

Figure 2a shows the results of Equation 1 computedfor the Voronoized networks. As the cutoff frequency forthe projector is varied (here it is lowered from 4Ωg), thecomputed Chern number converges to ν = −1 when allextended states in the top band lie above the cutoff fre-quency, confirming that the modes observed in Figure 1band c are topological in origin and predicting their direc-tion. The Chern number remains at its value of ν = −1for a broad range of frequencies in which any existingstates are localized, and thus do not contribute to theChern number. The Chern number returns to zero oncemore conductance-carrying extended states are includedin the calculation.

Having established this connection, we now discusshow the Chern number can be controlled. In particu-lar, we show that by considering alternative decorationsof the same initial point set, it is possible to flip the chi-rality of the edge modes or even provide multiple gaps

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FIG. 3. Alternative local decorations allow control of the edge mode chirality | (a-c), Kagomization of an arbitrarypoint set yields edge modes in gyroscopic networks with the opposite chirality to those in Voronoized networks. (d-e), Anotherlocal decoration of the initial point set allows for multiple gaps with either chirality. The amorphous ‘spindle’ network has twogaps with chiral edge modes: blue and red curves overlaying the density of states, D(ω), mark the excitation amplitude as afunction of frequency for the two cases. In (b-c), the spring frequency Ωk = k`2/Iω is set equal to the gravitational precessionfrequency, Ωg, while in (e), we chose Ωk = 7Ωg to broaden the lower (clockwise) mobility gap.

with differing chirality. One possible construction arisesnaturally from joining neighboring points in the origi-nal point set, leading to a Delaunay triangulation. Asshown in Supplementary Video 7, such networks showno gaps and no topology, suggesting that the local ge-ometry dictated by Voronoization is responsible for itsemergent topology. A clue can be found by noting thatthe Voronoized networks are locally akin to a honeycomblattice. The honeycomb is the simplest lattice with more

than one site per unit cell, a necessary condition for sup-porting a band gap in a lattice. Moreover, this lattice waspreviously found to be topological with the same Chernnumber [5].

Building on this insight, we introduce a second decora-tion, which we dub ‘kagomization,’ shown in Figure 3a.If applied to a triangular point set, Voronoization pro-duces a honeycomb lattice and kagomization produces akagome lattice, the simplest lattice with three sites per

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FIG. 4. Transition of a topological amorphous network to the trivial phase, and binary mixtures of Voronoizedand kagomized networks | a, Locally breaking inversion symmetry by increasing and decreasing the precession frequenciesof alternating gyroscopes competes with broken time reversal symmetry, triggering a transition to the trivial phase, with noedge modes. The precession frequency splitting, ∆, is tuned so that ΩA

g = Ωk(1 + ∆) and ΩBg = Ωk(1 −∆). b, Edge modes

are localized at the interfaces between kagomized and Voronoized networks, permitting sinuous channels for the propagationof unidirectional phonons. c, Excitations of a Voronoized region nested inside a kagomized network remain confined when theexcitation frequency is in a mobility gap unique to the kagomized network. d, When kagomized elements are randomly mixedinto a Voronoized network, the sign of the local, spatially-resolved Chern calculation is determined by the local geometry, withexcitations in a mobility gap biased toward the interface of the two clusters.

unit cell, which we have found to produce ν = +1 gyro-scopic metamaterials. Proceeding as with the Voronoizednetwork case (Figures 3b and c), we find the presenceof topologically protected modes with opposite directionand the corresponding opposite Chern number in the theband structure (see Supplementary Video 8). Other lo-cal constructions, such as the ‘spindle’ networks in Fig-ure 3d-e provide multiple mobility gaps, each with a dif-ferent edge mode chirality, offering a transmission direc-

tion tuned by frequency (see Supplementary Video 9).

One might think there could be a mapping from the ge-ometry of each vertex to the chirality of the edge modes.However, taken together, our Voronoized, kagomized,and spindle networks demonstrate that simply countingnearest neighbors is not sufficient to determine the topol-ogy: a description beyond nearest neighbors is required(see also Supplementary Information Figure 29). On theother hand, we are able to change the Chern number

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of a structure via local decorations. To uncover the ex-tent to which a network’s topology is stored locally, con-sider the projection operator, P . The projector value Pijmeasures the vibrational correlation between gyroscopej and gyroscope i when considering all modes above acutoff frequency. By explicitly computing its magnitudein our networks, we find that the magnitude of Pij fallsoff exponentially with distance (Supplementary Informa-tion Figure 11). Remarkably, explicitly cutting out asection of the network and embedding it in a networkwith a different spectrum results in only a slight changeto the local projector values (< 2%) (see SupplementaryInformation Figure 12). Since the Chern number is builtfrom these projector elements, it then follows that thelocal structure of the gyroscope network, combined withsome homogeneity of this local structure across the lat-tice, is all that is needed to determine the Chern number(c.f. Supplementary Video 10 and the SupplementaryInformation section entitled ‘Sign of Chern number fromnetwork geometry’).

This situation is reminiscent of electronic glasses inwhich the local binding structure gives rise to a local‘gap.’ Under weak assumptions of homogeneity, this gapcan be shown to extend to the whole system [19, 20]. Thecase with topology is similar: the next-nearest neighborangles in a network’s cell open a local ‘gap’ by breakingtime reversal symmetry.

For amorphous networks, we make the correspondencebetween the bulk topological invariant and the edgestates on the boundaries by considering a gyroscopic sam-ple shaped into an annulus (c.f. [25–27]). Adiabaticallytuning the interactions between pairs of gyroscopes alonga radial cut (by adding a fixture to one gyro from eachpair) pumps each edge mode into a neighboring mode,as shown in Supplementary Video 11 and in the Supple-mentary Information section ‘Spectral flow through adi-abatic pumping’. If we consider all states below a gapcutoff frequency, the process—which mimics the effect ofthreading a magnetic field through the centre of an an-nulus in an electronic system—trades one state localizedon the outer boundary for an extra state on the innerboundary of the annulus, which we connect to the real-space Chern number (Equation 1) in the SupplementaryInformation.

As in the lattice case, a mobility gap becomes topologi-cal due to time reversal symmetry breaking: bond anglesin these networks are not multiples of π/2 (c.f. [5]). Wecan probe this mechanism by eliminating a gap’s topol-ogy. Alternating the gravitational precession frequency,Ωg, of neighboring gyroscopes in a network mimics thebreaking of inversion symmetry on a local scale, an effectwhich competes against the time reversal gap openingmechanism. When the precession frequency differencebetween sites is large enough, this competing mechanismeliminates edge modes, triggering a transition to a ν = 0mobility gap, shown in Figure 4a and Supplementary

Videos 12 and 13.Equipped with these insights, we can easily engineer

networks which are heterogeneous mixtures of multiplelocal configurations. Figure 4b-d highlight some resultsof combining Voronoized and kagomized networks or en-capsulating one within another. Because the Voronoizedand kagomized networks share a mobility gap, excitationsare localized to their interface, offering a method of creat-ing robust unidirectional waveguides, such as the sinuouswaveguide shown in Figure 4b and Supplementary Video14. Figure 4c demonstrates that additional topologicalmobility gaps at higher frequency in the kagomized net-work allow bulk excitations to be confined to an encapsu-lated Voronoized region (see also Supplementary Video15). Random mixtures of the two decorations, shown inFigure 4d, demonstrate heterogeneous local Chern num-ber measurements (red for ν = +1 and blue for ν = −1),with mobility-gap excitations biased toward the inter-faces between red and blue regions (see SupplementaryInformation and Supplementary Video 16).

As our networks are structurally akin to liquids, theysupport topological modes in the absence of long rangespatial order. The details of the underlying point setare not essential, and neither are the details of the localVoronoization or kagomization procedures. We verifiedthis by replacing the centroidal construction [18] with aWigner-Seitz construction (see Supplementary Informa-tion Figures 24-26 for a comparison). Beyond mechan-ical materials, we find similar results in electronic tightbinding models of amorphous networks, underscoring thegenerality of the finding (Figure 32 of the SupplementaryInformation).

This study demonstrates that local interactions andlocal geometric arrangements are sufficient to gener-ate chiral edge modes, promising new avenues for en-gineering topological mechanical metamaterials gener-ated via imperfect self-assembly processes. Such self-assembled materials could be constructed, for instance,with micron-scale spinning magnetic particles. Since ourmethods bear substantial resemblance to tight-bindingmodels, our results also find direct application not onlyto electronic materials, as we have demonstrated, butalso to photonic topological insulators [11], acoustic res-onators [12, 13], and coupled circuits [28].

Acknowledgements: We thank Michael Levin,Charlie Kane, and Emil Prodan for useful discussions.This work was primarily supported by the Universityof Chicago Materials Research Science and EngineeringCenter, which is funded by National Science Foundationunder award number DMR-1420709. Additional supportwas provided by the Packard Foundation. The ChicagoMRSEC (U.S. NSF grant DMR 1420709) is also grate-fully acknowledged for access to its shared experimentalfacilities. This work was also supported by NSF EFRINewLAW grant 1741685.

Note added in proof: After the submission of this

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work, we became aware of a concurrent theoretical studyof the existence of amorphous electronic topological in-sulators in two and three dimensions [29].

Contributions: WTMI and NPM designed research.WTMI and AMT supervised research. NPM and LMNperformed the simulations and experiments. AMT,NPM, DH, and WTMI performed analytical calculations.DH contributed numerical tools. NPM and WTMI wrotethe manuscript. All authors interpreted the results andreviewed the manuscript.

Data Availability: The data that support the plotswithin this paper and other findings of this study areavailable from the corresponding authors upon request.

Methods: For small displacements (δX, δY ) of thepivot points from vertical alignment with the centers ofmass, the equation of motion for a gyroscope takes theform(

˙δXi

˙δYi

)≈ Ωg

(δYi−δXi

)+

Ωkk

NN∑j

(−Fij,YFij,X

), (2)

where Ωk = k`2/(Iω), and Ωg = `mg/(Iω), while k is thespring constant, ` is the length of the pendulum from thepivot to the center of mass, I is the moment of inertiaalong the spinning axis, ω is the angular spinning fre-quency, m is the mass of the gyroscope, and g is thegravitational acceleration. The simulations evolve Equa-tion 2 using a Runge-Kutta fourth-order explicit methodrun on a GPU using OpenCL.

To obtain spectra and normal modes, note that Equa-tion 2 defines the entries for a system’s dynamical matrix,D, such that

~ψ = D~ψ, (3)

where the components of ~ψ contain information on thedisplacements of the gyroscopes. In order to map to atight binding model, it is useful to write the displacementof the ith gyroscope as ψi ≡ δXi+ iδYi and note that theeigenvalue problem gives pairs of eigenvalues ±ω, so that

ψi = Lieiωt + Rie

−iωt ≡ ψLi + ψRi . (4)

Then

i

(ψLiψRi

)= Ωg

(−ψLiψRi

)− Ωk

2

∑j∈NN(i)

(ψLi − ψLj + ei2θij (ψRi − ψRj )−ψRi + ψRj − e−i2θij (ψLi − ψLj )

). (5)

The experiment in Figure 1c shows a chiral wave packetlocalized to the boundary in an experimental realizationof 195 gyroscopes in a Voronoized network based on apoint set generated from a jammed packing. Each gyro-scope consists of a spinning motor (∼ 150 Hz) housed ina 3D-printed enclosure (as in [5]), and each gyroscope is

suspended from an acrylic plate by a spring, an attach-ment method which was found to reduce damping.

To establish the equivalence of Equation 1 and the re-sponse to a point-like magnetic field, we study the ef-fect of a perturbation on the projection operator Pij ≡∑n∈band χn(i)χ∗n(j), where χ are the perturbed wave-

functions. In the Supplementary Information, we linkthe change in the diagonal elements to the charge accu-mulation near an applied magnetic field:

∆ρ = νe2

hBz, (6)

where ∆ρ is the change in charge density where the mag-netic field is applied, Bz is the magnetic field normal tothe sample, and ν is the Chern number of the occupiedbands when the sample is periodic.

Having shown that the Chern sum (Equation 1) isequal to the charge accumulated when a quantum of mag-netic flux is inserted, we can establish a correspondencebetween the bulk invariant and edge modes on the bound-ary by introducing a hole at the site of insertion. Thereal-space Chern number is then equal to the numberof edge states that accumulate on the inner boundaryas an effective magnetic flux is introduced through thehole [25–27]. The effective magnetic flux is manifest as aphase shift in the interactions for any loop of spring con-nections that encloses the hole. We construct this phaseshift by altering the subset of the nearest-neighbor inter-actions that traverse a cut of the annulus, such that theforce of one gyro on its neighbor across the cut is alteredby a rotation

F ∼ ψi − ψj → ψi − ψjeiθtwist . (7)

In the Supplementary Information, we propose a concretepicture of how this could be built in an experiment byattaching an extensible ring to a small number of gyro-scopes.

To see topological robustness in a simpler situation,we find similar behavior in an amorphous electronic tightbinding model using the model Hamiltonian

H = −t1∑〈ij〉

c†i cj − t2∑〈〈ij〉〉

e−iφijc†i cj , (8)

where 〈ij〉 denotes nearest neighbors ij and 〈〈ij〉〉 de-notes pairs of next-nearest neighbors (NNN). The pa-rameter t2 tunes the strength of all NNN hoppings, andφij controls the degree to which the hopping i→ j breakstime reversal symmetry (by tuning the imaginary term).As shown in the Supplementary Information, topologi-cal edge modes arise in amorphous tight binding lattices,whether the NNN hopping is uniform or bond angle-dependent.

∗ npmitchell@uchicago.edu; Corresponding author

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