Topological boundary modes in isostatic lattices

Abstract

Frames, or lattices consisting of mass points connected by rigid bonds or central-force springs, are important model constructs that have applications in such diverse fields as structural engineering, architecture and materials science. The difference between the number of bonds and the number of degrees of freedom in these lattices determines the number of their zero-frequency ‘floppy modes’. When these are balanced, the system is on the verge of mechanical instability and is termed isostatic. It has recently been shown that certain extended isostatic lattices exhibit floppy modes localized at their boundary. These boundary modes are insensitive to local perturbations, and seem to have a topological origin, reminiscent of the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators. Here, we establish the connection between the topological mechanical modes and the topological band theory of electronic systems, and we predict the existence of new topological bulk mechanical phases with distinct boundary modes. We introduce one- and two- dimensional model systems that exemplify this phenomenon.

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Figure 1: 1D SSH and isostatic lattice models.
Figure 2: Deformed kagome lattice model.
Figure 3: Zero modes at domain walls.
Figure 4: Zero modes at the edge.

References

  1. 1

    Phillips, J. C. Topology of covalent non-crystalline solids 2. medium-range order in chalcogenide alloys and α-Si((Ge). J. Non-Cryst. Solids 43, 37–77 (1981).

    ADS  Article  Google Scholar 

  2. 2

    Thorpe, M. F. Continuous deformations in random networks. J. Non-Cryst. Solids 57, 355–370 (1983).

    ADS  Article  Google Scholar 

  3. 3

    Feng, S. & Sen, P. N. Percolation on elastic networks—new exponent and threshold. Phys. Rev. Lett. 52, 216–219 (1984).

    ADS  Article  Google Scholar 

  4. 4

    Jacobs, D. J. & Thorpe, M. F. Generic rigidity percolation—the pebble game. Phys. Rev. Lett. 75, 4051–4054 (1995).

    ADS  Article  Google Scholar 

  5. 5

    Liu, A. J. & Nagel, S. R. Nonlinear dynamics—Jamming is not just cool any more. Nature 396, 21–22 (1998).

    ADS  Article  Google Scholar 

  6. 6

    Liu, A. J. & Nagel, S. R. Granular and jammed materials. Soft Matter 6, 2869–2870 (2010).

    ADS  Article  Google Scholar 

  7. 7

    Liu, A. J. & Nagel, S. R. The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1, 347–369 (2010).

    ADS  Article  Google Scholar 

  8. 8

    Torquato, S. & Stillinger, F. H. Jammed hard-particle packings: From Kepler to Bernal and beyond. Rev. Mod. Phys. 82, 2633–2672 (2010).

    ADS  Article  Google Scholar 

  9. 9

    Wyart, M., Nagel, S. R. & Witten, T. A. Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids. Europhys. Lett. 72, 486–492 (2005).

    ADS  Article  Google Scholar 

  10. 10

    Wyart, M. On the rigidity of amorphous solids. Ann. De Phys. 30, 1–96 (2005).

    Article  Google Scholar 

  11. 11

    Wilhelm, J. & Frey, E. Elasticity of stiff polymer networks. Phys. Rev. Lett. 91, 108103 (2003).

    ADS  Article  Google Scholar 

  12. 12

    Heussinger, C. & Frey, E. Floppy modes and nonaffine deformations in random fibre networks. Phys. Rev. Lett. 97, 105501 (2006).

    ADS  Article  Google Scholar 

  13. 13

    Huisman, L. & Lubensky, T. C. Internal stresses, normal modes and non-affinity in three-dimensional biopolymer networks. Phys. Rev. Lett. 106, 088301 (2011).

    ADS  Article  Google Scholar 

  14. 14

    Broedersz, C., Mao, X., Lubensky, T. C. & MacKintosh, F. C. Criticality and isostaticity in fibre networks. Nature Phys. 7, 983–988 (2011).

    ADS  Article  Google Scholar 

  15. 15

    Souslov, A., Liu, A. J. & Lubensky, T. C. Elasticity and response in nearly isostatic periodic lattices. Phys. Rev. Lett. 103, 205503 (2009).

    ADS  Article  Google Scholar 

  16. 16

    Mao, X. M., Xu, N. & Lubensky, T. C. Soft modes and elasticity of nearly isostatic lattices: Randomness and dissipation. Phys. Rev. Lett. 104, 085504 (2010).

    ADS  Article  Google Scholar 

  17. 17

    Mao, X. M. & Lubensky, T. C. Coherent potential approximation of random nearly isostatic kagome lattice. Phys. Rev. E 83, 011111 (2011).

    ADS  Article  Google Scholar 

  18. 18

    Mao, X. M., Stenull, O. & Lubensky, T. C. Elasticity of a filamentous kagome lattice. Phys. Rev. E 87, 042602 (2013).

    ADS  Article  Google Scholar 

  19. 19

    Kapko, V., Treacy, M. M. J., Thorpe, M. F. & Guest, S. D. On the collapse of locally isostatic networks. Proc. R. Soc. A 465, 3517–3530 (2009).

    ADS  MathSciNet  Article  Google Scholar 

  20. 20

    Maxwell, J. C. On the calculaton of the equilibrium stiffness of frames. Phil. Mag. 27, 294–299 (1865).

    Article  Google Scholar 

  21. 21

    Calladine, C. R. Buckminster Fuller’s ‘tensegrity’ structures and clerk Maxwell’s rules for the construction of stiff frames. Int. J. Solids Struct. 14, 161–172 (1978).

    Article  Google Scholar 

  22. 22

    Sun, K., Mao, X. & Lubensky, T. C. Surface phonons, elastic response, and conformal invariance in twisted kagome lattices. Proc. Natl Acad. Sci. USA 109, 12369–12374 (2012).

    ADS  Article  Google Scholar 

  23. 23

    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).

    ADS  Article  Google Scholar 

  24. 24

    Haldane, F. D. M. Model for a quantum hall effect without landau levels—condensed matter realization of the parity anomaly. Phys. Rev. Lett. 61, 2015–2018 (1988).

    ADS  MathSciNet  Article  Google Scholar 

  25. 25

    Kane, C. L. & Mele, E. J. Z(2) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    ADS  Article  Google Scholar 

  26. 26

    Bernevig, B. A., Hughes, T. L. & Zhang, S-C. Quantum spin hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    ADS  Article  Google Scholar 

  27. 27

    Moore, J. E. & Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306 (2007).

    ADS  Article  Google Scholar 

  28. 28

    Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

    ADS  Article  Google Scholar 

  29. 29

    Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Article  Google Scholar 

  30. 30

    Qi, X-L. & Zhang, S-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    ADS  Article  Google Scholar 

  31. 31

    Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetalene. Phys. Rev. Lett. 42, 1698 (1979).

    ADS  Article  Google Scholar 

  32. 32

    Nakahara, M. Geometry, Topology and Physics (Hilger, 1990).

    Google Scholar 

  33. 33

    Dirac, P. A. M. The quantum theory of the electron. R. Soc. Lond. Proc. A 117, 610–624 (1928).

    ADS  Article  Google Scholar 

  34. 34

    Witten, E. Dynamical breaking of supersymmetry. Nucl. Phys. B 188, 513–554 (1981).

    ADS  Article  Google Scholar 

  35. 35

    Cooper, F., Khare, A. & Sukhatme, U. Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–385 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  36. 36

    Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).

    ADS  Article  Google Scholar 

  37. 37

    Jackiw, R. & Rebbi, C. Solitons with fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976).

    ADS  MathSciNet  Article  Google Scholar 

  38. 38

    Volovik, G. E. The Universe in a Helium Droplet (Clarenden, 2003).

    Google Scholar 

  39. 39

    Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    ADS  Article  Google Scholar 

  40. 40

    Wang, Z., Chong, Y. D., Joannopoulos, J. D. & Soljačić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008).

    ADS  Article  Google Scholar 

  41. 41

    Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009).

    ADS  Article  Google Scholar 

  42. 42

    Berg, N., Joel, K., Koolyk, M. & Prodan, E. Topological phonon modes in filamentary structures. Phys. Rev. E 83, 021913 (2011).

    ADS  Article  Google Scholar 

  43. 43

    Callias, C. Axial anomalies and index theorems on open spaces. Commun. Math. Phys. 62, 213–234 (1978).

    ADS  MathSciNet  Article  Google Scholar 

  44. 44

    Bott, R. & Seeley, R. Some remarks on paper of callias. Commun. Math. Phys. 62, 235–245 (1978).

    ADS  Article  Google Scholar 

  45. 45

    Hirayama, M. & Torii, T. fermion fractionalization and index theorem. Prog. Theor. Phys. 68, 1354–1364 (1982).

    ADS  MathSciNet  Article  Google Scholar 

  46. 46

    Niemi, A. J. & Semenoff, G. W. fermion number fractionalization in quantum field theory. Phys. Rep. 135, 99–193 (1986).

    ADS  MathSciNet  Article  Google Scholar 

  47. 47

    Lakes, R. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987).

    ADS  Article  Google Scholar 

  48. 48

    Lawler, M. J. Emergent gauge dynamics of highly frustrated magnets. New J. Phys. 15, 043043 (2013).

    ADS  Article  Google Scholar 

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Acknowledgements

T.C.L. is grateful for the hospitality of the Newton Institute, where some of this work was carried out. This work was supported in part by a Simons Investigator award to C.L.K. from the Simons Foundation and by the National Science Foundation under DMR-1104707 (T.C.L.) and DMR-0906175 (C.L.K.).

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C.L.K. and T.C.L. contributed to the formulation of the problem, theoretical calculations, and the preparation of the manuscript.

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Correspondence to T. C. Lubensky.

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Kane, C., Lubensky, T. Topological boundary modes in isostatic lattices. Nature Phys 10, 39–45 (2014). https://doi.org/10.1038/nphys2835

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