Annual Report 2020

Leveraging Symmetries to Invent Exotic Materials

Mathematics and Physical Sciences

For the past two decades the field of metamaterials — the design and fabrication of materials that have properties not found in nature — has been an exciting area of research in physics and engineering. The Simons Collaboration on Extreme Wave Phenomena Based on Symmetries is part of this line of inquiry. Researchers in the collaboration are interested in discovering, exploring and creating materials with exotic properties that will interact with electromagnetic and acoustic waves in unusual and desirable ways. These types of properties are created by leveraging various kinds of symmetries and ‘symmetry breaking’ in the engineered materials. 

The word ‘symmetry’ often brings to mind geometrical features: a square, which looks the same when it is rotated 90 degrees or reflected across horizontal, vertical or diagonal lines; a circle, which lines up with itself when rotated by any angle or reflected across any diameter line; or a snowflake, which looks unchanged when rotated by 60 degrees or reflected across one of many lines of bilateral symmetry. 

More broadly, a symmetry is a general transformation of an object that, when performed, leaves the object in a state indistinguishable from its initial state. In addition to the more familiar geometrical symmetries, researchers in the collaboration are studying symmetries in three other general classes: unfolding symmetries, dynamical symmetries, and supersymmetries and dualities. Unfolding or scaled symmetries occur in materials whose wave equations are governed by fractal behavior. The dynamical class refers to symmetries that are based on the evolution of properties of the material.

The Simons Collaboration on Extreme Wave Phenomena Based on Symmetries is exploring and blending four broad symmetry classes: geometrical symmetries, unfolding symmetries, dynamical symmetries, and supersymmetries and dualities. Credit: Lucy Reading-Ikkanda/Simons Foundation

Supersymmetries and dualities, in theoretical physics and mathematics, are phenomena in which seemingly unrelated systems exhibit the same responses. Postdoc Michel Fruchart and his supervisor Vincenzo Vitelli hit upon the potential for using these ideas to generate exotic material properties almost by chance while playing with Lego structures a few years ago. Their tantalizing discovery spurred new research into the physics of dualities in the context of wave phenomena. Vitelli, a principal investigator of the new collaboration and physics professor at the University of Chicago, says, “We just stumbled into this effect, and then we realized that it’s a smoking gun for a more general mathematical formalism that occurs in a variety of mechanical, optical and electronic systems.”

Researchers are looking at both the properties endowed by particular symmetries in a metamaterial and the exotic behavior that can be created when symmetry is broken. “For me, it’s exciting to think about how to design a structure so that I can break symmetry in a specific way or add symmetry-breaking for functionality,” says Katia Bertoldi, a principal investigator of the collaboration and engineering professor at Harvard University.

Besides exploring different kinds of symmetry, the researchers are also excited by the potential for combining several types of symmetries in the same metamaterial to control the overall response. “Can we combine different aspects of symmetry to come up with something that’s more than the sum of the individual parts?” asks Nader Engheta, a principal investigator and professor in both the electrical and systems engineering department and the physics and astronomy department at the University of Pennsylvania. “That could open the door to a lot of interesting future devices and possibilities.”

A twisted metasurface bilayer enables the detection of chiral molecules (which have ‘right-handed’ or ‘left-handed’ mirror symmetry) at very low concentrations. Credit: Y. Zhao et al./Nature Communications 2017

For example, although different types of waves seem very different in day-to-day life — sound and light, for example, are perceived in different ways and have different uses — many of the same principles apply to the study of all waves. “If you look at the problems from a theoretical perspective, they’re not that different,” says Andrea Alù, collaboration director and Einstein professor of physics at the City University of New York (CUNY) Graduate Center, as well as founding director of the Photonics Initiative at the CUNY Advanced Science Research Center.

For that reason, researchers often build models and devices that work with one kind of wave before trying them in a different setting. “The beauty of this field is that we have a lot of tools at our disposal,” says Demetrios Christodoulides, a principal investigator of the collaboration and professor of optics at the University of Central Florida. “We come up with a theoretical discovery, and then we actually have the flexibility to decide how best to demonstrate it.” For example, researchers might first fabricate structures that conduct acoustic waves: Because these waves are relatively slow, such structures tend to be large-scale and simple to design. Later, tests of the concepts can be translated to the more complicated microscopic worlds of optical or electromagnetic devices.

Scientists in the collaboration can fabricate prototypes of new materials in a matter of weeks, but perfecting these designs and then applying them in real-world devices of course takes much longer. Nevertheless, researchers are hopeful that their materials could eventually be used to improve technologies as disparate as medical imaging, optical computing and cellular communication networks.

Although the collaboration officially began only in September 2020, it is building on existing research collaborations among several of its principal investigators and their research groups. “The goal of the collaboration is to leverage all these initial efforts that our team members have pioneered, using symmetries to guide the optimal designs of metamaterials for various technologies, and bring them together, connecting the dots to build a unified theory that can enable us to discover new materials and new functionalities for many technologies,” Alù says. These earlier projects have allowed investigators to get the collaboration up and running quickly in the short time since their official launch date.

Researchers involved in the collaboration come from a wide range of academic backgrounds. “It’s interesting to get perspectives from mathematics, from physics, from engineering, and try to find common ground,” Bertoldi says. “It always takes some time to make sure that we understand each other, but that’s the fun part.”

As is often the case with research that spans theoretical and applied science, inspiration does not flow in one direction, from theory to applications. Instead, “it’s a back-and-forth,” Christodoulides says. “We have a problem in mind and then we see what kinds of tools we can bring to address this problem. At the same time, we cannot address a problem unless we really know our toolbox.” Insights from the theoretical and applied aspects of the collaboration create a complex cycle of gradual progress on all fronts.

“The Simons Foundation gives us the opportunity not only to continue existing collaborations, but also to expand them more broadly,” Engheta notes. The new structure will be especially valuable when it comes to cross-disciplinary projects, which can be more difficult in traditional academic collaborations. “When we study a phenomenon, that phenomenon does not just go into a box called physics or a box called mathematics,” Engheta says. “It’s a combination of everything.”