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Topological boundary modes in isostatic lattices


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.


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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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The authors declare no competing financial interests.

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Kane, C., Lubensky, T. Topological boundary modes in isostatic lattices. Nature Phys 10, 39–45 (2014).

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