Amorphous topological insulators constructed from random point sets

Abstract

The discovery that the band structure of electronic insulators may be topologically non-trivial has revealed distinct phases of electronic matter with novel properties1,2. Recently, mechanical lattices have been found to have similarly rich structure in their phononic excitations3,4, giving rise to protected unidirectional edge modes5,6,7. In all of these cases, however, as well as in other topological metamaterials3,8, the underlying structure was finely tuned, be it through periodicity, quasi-periodicity or isostaticity. Here we show that amorphous Chern insulators can be readily constructed from arbitrary underlying structures, including hyperuniform, jammed, quasi-crystalline 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 real-space 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 self-assembly of non-crystalline topological metamaterials on the micro and macro scale.

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Fig. 1: Local structure gives rise to chiral edge modes.
Fig. 2: Chern number calculations confirm topological mobility gaps.
Fig. 3: Alternative local decorations allow control of the edge mode chirality.
Fig. 4: Transition of a topological amorphous network to the trivial phase, and binary mixtures of Voronoized and kagomized networks.

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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 DMR-1420709. 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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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.

Corresponding authors

Correspondence to Noah P. Mitchell or William T. M. Irvine.

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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 real-space 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

Real-space 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 frequency-dependent 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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Mitchell, N.P., Nash, L.M., Hexner, D. et al. Amorphous topological insulators constructed from random point sets. Nature Phys 14, 380–385 (2018). https://doi.org/10.1038/s41567-017-0024-5

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