# Delving Into the Mysteries of Waves

## Simons Collaboration on Localization of Waves

In the late 1950s, future Nobel Prize-winning physicist Philip Anderson discovered a puzzling phenomenon: In theory, the conductivity of materials could be reduced and even vanish when a sufficient amount of disorder was introduced into the material. Instead of flowing along straight paths, electrons would remain trapped at specific locations. It took decades before this phenomenon, called Anderson localization, was observed in real experiments, and scientists are still trying to understand exactly how and why it occurs.

Anderson localization is just one instance of a broader phenomenon known as wave localization. Wave localization can happen in almost any material in which waves or vibrations carry energy or particles, which includes everything from sound waves to light waves, from vibrations of molecules to quantum properties of atoms or electrons in matter. Waves are said to be localized when they are confined to small areas due to the structure of the ambient medium. Researchers in the Simons Collaboration on Localization of Waves are working to put localization on a mathematically and physically rigorous footing, which could eventually help researchers harness it to create materials with desirable properties.

The roots of the collaboration extend back to 2009, when collaboration principal investigator Marcel Filoche, a physicist at the École Polytechnique, met Svitlana Mayboroda, a mathematician at the University of Minnesota and now also director of the collaboration. Filoche was working on sound localization and absorption caused by geometrical disorder, while Mayboroda was interested in the properties of steady-state solutions to partial differential equations. Soon they realized that their respective projects were deeply connected, and together they began to develop a general theory of wave localization. Their early partnership would eventually expand to become the Simons Collaboration on Localization of Waves, launched in 2018.

The collaboration now has 10 PIs — five mathematicians and five physicists — and dozens of students, postdocs and affiliated scientists from vastly different backgrounds. Participants include mathematicians from the fields of harmonic analysis and partial differential equations (led by Guy David, David Jerison and Yves Meyer), along with experimental physicists working in quantum optics (led by Alain Aspect), semiconductors (led by James Speck and Claude Weisbuch) and organic semiconductors (led by Richard Friend), and specialists in computation (led by Douglas Arnold) to tie it all together.

The cross-pollination between different areas of research has already led to rich cycles of interdisciplinary insights. Physics inspires mathematics, mathematical discoveries spur physical experiments and computational models both validate prior conjectures and lead to new avenues of exploration. “It is absolutely magical,” Mayboroda says. “We created this fantastic team of people who are really the best in each direction of research.”

The central puzzle of localization is how even a small amount of disorder can effectively hold waves in place. It is not surprising that disorder decreases conductivity. In perfectly ordered materials such as crystals, waves travel in predictable, straight paths, like light rays in the air. Also predictably, disorder interrupts those paths, but the extent to which disorder can cause waves to localize seems disproportionately great. “The physics and mathematics of the previous century is very well equipped to deal with ordered structures,” Mayboroda says. “But localization in particular — and the world in general — is run by disorder.” The collaboration researchers’ challenge is to find the order underlying systems that seem completely disordered so that they can describe and predict them.

The traditional approach to studying disorder-induced localization is probabilistic and statistical in nature. It gives a picture of the most likely behaviors of waves in disordered media, but it does not provide a strong theoretical framework for predicting where and how localization works. The collaboration aims to develop a rigorous mathematical understanding of localization. “We found the same mathematical objects popping up in very different fields,” Filoche says, from the subatomic waves of quantum mechanics to the much larger ones in acoustics. “It seemed like maybe there was a more universal scheme at work.”

Indeed, the original inspiration of Mayboroda and Filoche came from an unexpected area: the study of vibrating plates. Great 19th-century mathematicians, including Ernst Chladni and Sophie Germain, had investigated how waves form on thin plates when different kinds of vibrations are applied. Mayboroda and Filoche, looking at the same problem, discovered that they could dramatically alter the propagation of vibrations by blocking just one point in the plate, thus localizing waves at specific frequencies. They developed a new theoretical tool, called the localization landscape, to describe and predict this phenomenon.

That work, published in 2012, opened the door to an expansion of the research group, in the direction of both mathematics and physics. Mathematicians Douglas Arnold from the University of Minnesota, Guy David from the Université Paris-Saclay, and David Jerison from the Massachusetts Institute of Technology joined the team; together, their work led to the discovery of ‘effective potential,’ which plays a key role in predicting the quantum energies and densities of states in disordered systems.

At the same time, this theory was immediately applied successfully in semiconductor physics, where Claude Weisbuch, of the École Polytechnique and the University of Santa Barbara, realized that the landscape approach could help physicists understand how localization occurs in light-emitting diodes (LED) and photovoltaic cells. The disorder present in modern LED materials traps the electronic quantum waves, forcing them to concentrate along specific paths and at specific locations and to emit photons. Greater theoretical understanding of localization could help researchers improve the efficiency of LEDs and solve several vexing puzzles, such as why green LEDs are much less energy-efficient than red and blue ones. The collaboration hopes the landscape approach may help solve what researchers refer to as the ‘green gap.’

By the time the group applied for Simons Foundation funding, experimental physicists such as Alain Aspect of the Institut d’Optique in Paris, who studies small systems of atoms at ultralow temperatures, and Richard Friend of the University of Cambridge, who uses high-precision lasers to observe localization at the nanoscale in organic semiconductors, had joined the team, together with Yves Meyer, a mathematician at the École Normale Supérieure Paris-Saclay and an expert in harmonic analysis.

The collaboration is relatively young, but members have already succeeded in building on Filoche and Mayboroda’s localization landscape. A 2019 result of Filoche, Mayboroda and David described the ‘landscape law,’ a breakthrough in understanding the energy levels of localized waves in a mathematically rigorous way.

Mathematics and theoretical physics research sometimes have surprising applications that are only discovered and exploited centuries after the theory is developed. For localization, the timeline is proving much shorter. The theory has already found applications in LEDs and in protein vibrations, and it is about to be developed for solar cells and even for quantum computing. “It is absolutely funny, and a big lesson, that things that were tested on vibrating plates are theoretical tools that could be of help to build the next quantum devices,” Filoche says.