Topological boundary modes in isostatic lattices

By C. L. Kane, Tom C. Lubensky. Nature Physics, 2013

794 citations recorded by the source corpus.

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. The mathematical connection between isostatic lattices—which are relevant for granular matter, glasses and other ‘soft’ systems—and topological quantum matter is as deep as it is unexpected.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/95965139487e41ad6622336ad657ba8a3c2e42f9

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In Atomic and Molecular Physics, and Optics

In Materials Chemistry

Citations in the corpus

Architecture and coevolution of allosteric materials (2016, background): “Recent works have shown that marginally connected crystals can display edge modes, leading to exponentially growing response when displacement are imposed at the boundary of the system [27, 28, 29].”
Topological photonics (2014, background): “Moreover, the concepts and realizations of topological photonics can be translated to other bosonic systems like surface plasmons, excitons [82], excitonpolaritons [83, 84], phonons [85, 86] and magnons [87].”
Erratum: On variational arguments for vibrational modes near jamming (2016, background): “These networks can present interesting topological properties [15,16,32].”
A review on application of mechanical metamaterials for vibration control (2021, background): “In this review, the architectures mentioned can get mapped onto effective parameters because the lattice constant taken is much smaller than the wavelength’s characteristic length [21, 22].”
Reciprocal skin effect and its realization in a topolectrical circuit (2019, background): “[27] C.”
Intrinsically Polar Elastic Metamaterials (2017, methodology): “In the theory of Kane and Lubensky [8, 18], isostatic frames are described by a polarization vector RT akin the electric polarization of dielectrics.”
Amorphous topological insulators constructed from random point sets (2016, background): “ical lattices have been found to have similarly rich structure in their phononic excitations [3, 4], giving rise to protected uni-directional edge modes [5–7].”