Physics Colloquium: Tom Lubensky, "Topological Mechanics of Critically Coordinated Lattices"
Topological Mechanics of Critically Coordinated Lattices
Tom Lubensky Emeritus Christopher H. Browne Distinguished Professor of Physics
Department of Physics and Astronomy
University of Pennsylvania
Frames consisting of nodes connected pairwise by rigid rods or central-force springs, possibly with preferred relative angles controlled by bending forces, are useful models for systems as diverse as architectural structures, crystalline and amorphous solids, sphere packings and granular matter, networks of semi-flexible polymers, proteins, and origami. Particularly interesting today is the increasing number of 3D-printed micron-scale metamaterials. This talk will present an overview of the elastic and vibrational properties of versions of these frames, called Maxwell lattices, whose constraints match the translational degrees of freedom of their nodes. They include the square, kagome, and pyrocholore and lattices and their modifications with nearest-neighbor central-force springs as well as jammed packings of soft spheres. Like the Su-Schrieffer-Heeger model of Polyacetylene, topological insulators, and Weyl semi-metals, these lattices have a topological characterization that in their case determines the number and nature of their zero-energy edge modes, the nature of their long-wavelength elasticity, and whether or not they have isolated topologically protected Weyl-like zero modes in the bulk. If time permits, the talk will present a mechanical model whose vibrational spectrum reproduces the electronic spectrum of graphene with different hopping for each of the three bond directions.
- K. Sun, A. Souslov, X. M. Mao, and T.C. Lubensky, PNAS 109, 12369-12374 (2012).
- C.L. Kane and T.C. Lubensky, Nature Physics 10, 39-45 (2014)
- T. C. Lubensky, C. L. Kane, X. Mao, A. Souslov, K. Sun, Rep. Prog. Phys. 78, 073901 (2015).
- D. Z. Rocklin, B. G. G. Chen, M. Falk, V. Vitelli, and T. C. Lubensky, "Mechanical Weyl Modes in Topological Maxwell Lattices," Phys. Rev. Lett. 116, 135503 (2016).
- O. Stenull, C. L. Kane, and T. C. Lubensky, "Topological Phonons and Weyl Lines in Three Dimensions," Phys. Rev. Lett. 117 (6), 068001 (2016).  J. E. S. Socolar, T. C. Lubensky, and C. L. Kane, "Mechanical
- graphene," New Journal of Physics 19, 025003 (2017).
My research focuses on "soft" materials such as liquid crystals, membranes, vesicles, Langmuir films, and the many realizations of complex fluids such as microemulsions. My approach to the study of these materials is phenomenological: their properties at length scales several times molecular lengths can be described by effective free energies and by hydrodynamical equations, which depend only on the symmetry and conservation laws of their equilibrium phases. Associated with each thermodynamic phase are elastic rigidities, low-frequency hydrodynamics modes, and topological defects that are collectively responsible for most of the remarkable properties of soft materials. Thus, to understand the properties of any given phase (such as for example, the lamellar phase of a lyotropic liquid crystal), one needs to identify broken symmetries and conservation laws to determine harmonic long-wavelength elasticity and hydrodynamics. Then, one needs to investigate the effects of thermal fluctuations on the naive harmonic theories. Thermal fluctuations are almost by definition strong in soft systems, and they can lead to significant modifications of harmonics theories, which are best studied using field theories and the renormalization group. I have used these techniques to predict a new phase of matter in chiral liquid crystals in which there is a regular array of twist grain boundaries separating layered smectic-A slabs, whose layer normal rotates in a helical fashion along a pitch axis parallel to the plane of the layers. This twist-grain-boundary (TGB) phase is the analog in liquid crystals of the Abrikosov vortex lattice phase in superconductors. I have studied the fluctuations of two-dimensional "fishnet" solids and found that coupling to the capillary or height mode leads to an anomalous elasticity in which the long wavelength bulk and shear moduli renormalize to zero. I have also investigated topological defects and topologically induced interactions in liquid crystalline colloids and emulsions, the microscopic origin of chiral interactions in liquid crystals, chiral phases in polymeric and discotic liquid crystals, and interactions between colloidal particles dispersed in DNA solutions. My recent research activities include investigations of new "sliding phases" of matter, microrheology, liquid crystalline elastomers, and flow of granular material. Sliding phases occur in 3D stacks of two-dimensional systems, such as xy-models, crystals, or 2D smectics, with a continuous symmetry. They behave like two-dimensional systems with power-law decay of correlations even though they are coupled in three dimensions. Microrheology is a technique that measures the complex shear moduli of viscoelastic media from the response and fluctuations of colloidal beads dispersed in them. It can be used on smaller samples than, and it provides information at higher frequency than standard rheological techniques. Nematic elastomers combine the elastic properties of rubber with the orientational order of a nematic liquid crystal. They exhibit a remarkable soft elasticity, brought about by a spontaneous broken symmetry, in which a shear modulus measuring the energy of strains in planes containing the anisotropy axis vanishes. Granular material can flow like a fluid when it is subjected to a sufficiently large shear. I have developed hydrodynamics theories to describe this flow including the flow of a two-dimensional gas of chiral rattlebacks.