Stretching Boundaries

Tom Lubensky and Charles Kane combine two distinct areas of physics in new research.
January 1, 2014

Topological insulators and isostatic lattices make strange bedfellows—or at least they seemed to until Tom Lubensky, Christopher H. Browne Distinguished Professor of Physics, started bugging his colleague Charles Kane about their similarities. Their eventual collaboration led to the recently published “Topological Boundary Modes in Isostatic Lattices” in Nature Physics.

“Tom and I have written a couple of papers together over the years. We talk a lot with each other and throw ideas around,” says Kane, Class of 1965 Term Professor of Physics and Astronomy. “We each come at problems from different points of view, and I think we have different tastes, which makes the result more interesting.”

So what exactly are topological insulators and isostatic lattices, and how do they intersect? In quantum condensed matter physics, a topological insulator is a material that doesn’t conduct electricity on its interior. However, a special property of its interior guarantees that it does conduct on its surface. The takeaway is that something is happening on the boundary of the material that’s not happening inside it.

As for isostatic lattices, examples can be found in soft condensed matter physics as well as in architecture. They are useful for designing bridges because in their mechanically stable form they are free to translate and rotate rigidly without breaking.

“Consider three rods confined to a two-dimensional plane,” says Lubensky. “They can each translate freely in two directions and rotate freely about their centers for a total of nine zero-energy motions. If you begin to connect the rods it decreases their ability to translate and therefore their energy motions. Once they are connected at their ends to form a triangle, they become a mechanically stable lattice you might use in architecture.”

These lattices’ significance to the combined work of both professors is found once they begin to distort and create “floppy modes,” a pathway for collapse.

“What Professor Lubensky realized is that there are certain ways you can deform lattices so that the floppy modes appear on the boundaries,” says Kane. “So this existence of these on the edges of the lattices smells similar to the phenomenon of topological insulators, in that something is happening on the boundary that’s not happening in the interior.”

Combining these phenomena holds the potential for the creation of new classes of metamaterials. Metamaterials are composed of numerous smaller pieces, typically nanometer or micro-sized, and designed to collectively have some special electromagnetic or mechanical property. A famous example is the so-called electromagnetic invisibility cloak, which blocks reflections, therefore hiding the object it’s covering. Mechanical metamaterials based on isostatic lattices can be engineered to have what is called a negative Poisson ratio: When stretched along one direction, they expand rather than contract along perpendicular directions.

“We’re in the conceptual stage at the moment,” says Lubensky. “It remains to be seen exactly how far the community can take this.”