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.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Haldane, F. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).
Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2013).
Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009).
Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015).
Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015).
Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).
Sussman, D. M., Stenull, O. & Lubensky, T. C. Topological boundary modes in jammed matter. Soft Matter 12, 6079–6087 (2016).
Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, Cambridge, 2000).
Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).
Rechtsman, M. C. et al. Photonic floquet topological insulators. Nature 496, 196–200 (2013).
Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).
Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alu, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).
Meeussen, A. S., Paulose, J. & Vitelli, V. Geared topological metamaterials with tunable mechanical stability. Phys. Rev. X 6, 041029 (2016).
Thouless, D. J. Wannier functions for magnetic sub-bands. J. Phys. C 17, L325–L327 (1984).
Huo, Y. & Bhatt, R. N. Current carrying states in the lowest Landau level. Phys. Rev. Lett. 68, 1375–1378 (1992).
Thonhauser, T. & Vanderbilt, D. Insulator/Chern-insulator transition in the Haldane model. Phys. Rev. B 74, 235111 (2006).
Florescu, M., Torquato, S. & Steinhardt, P. J. Designer disordered materials with large, complete photonic band gaps. Proc. Natl Acad. Sci. USA 106, 20658–20663 (2009).
Weaire, D. & Thorpe, M. F. Electronic properties of an amorphous solid. I. A simple tight-binding theory. Phys. Rev. B 4, 2508–2520 (1971).
Weaire, D. Existence of a gap in the electronic density of states of a tetrahedrally bonded solid of arbitrary structure. Phys. Rev. Lett. 26, 1541–1543 (1971).
Haydock, R., Heine, V. & Kelly, M. J. Electronic structure based on the local atomic environment for tight-binding bands. J. Phys. C 5, 2845–2858 (1972).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
Prodan, E. Non-commutative tools for topological insulators. New. J. Phys. 12, 065003 (2010).
Bianco, R. & Resta, R. Mapping topological order in coordinate space. Phys. Rev. B 84, 241106 (2011).
Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632–5633 (1981).
Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982).
Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).
Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).
Agarwala, A. & Shenoy, V. B. Topological insulators in amorphous systems. Phys. Rev. Lett. 118, 236402 (2017).
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.
Author information
Authors and Affiliations
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
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567-017-0024-5
This article is cited by
-
Observation of spin-momentum locked surface states in amorphous Bi2Se3
Nature Materials (2023)
-
Top-down patterning of topological surface and edge states using a focused ion beam
Nature Communications (2023)
-
An exact chiral amorphous spin liquid
Nature Communications (2023)
-
Observation of novel topological states in hyperbolic lattices
Nature Communications (2022)
-
Engineered disorder in CO2 photocatalysis
Nature Communications (2022)