Simons Collaboration on Localization of Waves Annual Meeting 2022

The third annual meeting of the Localization of Waves Collaboration sponsored by the Simons Foundation hosted 157 world-class scientists (67 in person and 90 remote) in New York City to explore profound connections between disorder, geometric complexity, the behavior of waves, and to discuss recent advances in mathematical analysis and physics of wave localization.

The topics of the meeting were centered around the results and future plans of the collaboration. Collaboration Director Svitlana Mayboroda (University of Minnesota) led with the first presentation, “Localization of Waves: Director’s Overview,” in which she discussed the structure of the project and provided an overview of the key achievements and principal challenges across all the involved disciplines. Founding PI James Speck’s (University of California, Santa Barbara) talk “Unraveling the Inner Working of GaN-Based Light-Emitting Diodes” concentrated on the theoretical and experimental research on InGaN semiconductors, as well as their potential impact on the projected energy savings from lighting. The talk of the Nobel Prize winner William D. Phillips (University of Maryland), “The Coldest Stuff in the Universe: From Quantum Clocks to Quantum Simulators,” introduced the audience to some ideas behind the experiments with cold atoms systems, from their original conception to the most recent results pertaining to the observations of Anderson localization. Founding PI Guy David (Université Paris-Saclay) presented the landscape law for the integrated density of states and estimates for the exponential decay of the eigenfunctions using the geodesic distance associated to the landscape in his talk, “Estimates on Eigenfunctions and Eigenvalues Using the Landscape Function.” Closing out the first day of presentations, Founding PI Sir Richard Friend’s (University of Cambridge) talk, “Static and Dynamic Disorder in Organic and Perovskite Semiconductors,” led to discussions on the role of localization and the use of the localization landscape for organic semiconductors.

The second day of the conference began with a talk by Camillo De Lellis (Institute for Advanced Study), entitled “The Mumford-Shah Functional and Its Challenges,” and was devoted to the free boundaries and minimal surfaces. The next talk, “The Wigner-Weyl approach to wave localization,” was presented by Founding PI Marcel Filoche (École Polytechnique) and surveyed the recent advances using the localization landscape in the phase space with applications to the predictions of spectral functions, light absorption in disordered alloys and, potentially, the prediction of the mobility edge of Anderson localization. The conference concluded with the collaboration’s most recent PI Hugo Duminil-Copin’s (Université de Genève) talk “On the Geometry of Nodal Lines of Random Waves.” It brought together the challenges of understanding the topology of monochromatic random waves and recent advances in percolation theory.

Ten posters, representing collaborative subgroup teams, were on display throughout the meeting to showcase recent advancements:

• Electronic structure of mixed halide perovskites
  • Yun Liu, University of Cambridge
• Estimates for the Green function
  • Linhan Li, University of Minnesota
• Evidence of localization effect on electron transport induced by alloy disorder in InGaN by low energy photoemission spectroscopy
  • Claude Weisbuch, University of California at Santa Barbara
• Improved Weyl’s law approximation to IDOS
  • Wei Wang, University of Minnesota
• Intrinsic Lipschitz graphs in Carnot groups
  • Antoine Julia, Université Paris Saclay
• The landscape function and the critical radius function: New applications for magnetic Schrödinger operators
  • Bruno Poggi, Universitat Autònoma de Barcelona
• Localization landscape for interacting Bose gases in one-dimensional speckle potentials
  • Filippo Stellin, École Normale Supérieure Paris-Saclay
• Non-local distances and geometry of measures
  • Cole Jeznach, University of Minnesota
• Ultracold atoms in strong disorder: towards the Anderson transition
  • Xudong Yu, Laboratoire Charles Fabry, IOGS, Université Paris-Saclay
• Wigner-Weyl description of light absorption in correlated semiconductor alloys using the localization landscape theory
  • Jean-Philippe Banon, Ecole Polytechnique

The annual meeting helped solidify future goals and objectives for the collaboration.

• Svitlana Mayboroda will extend her work on developing a new calculus to account for the structures recently unveiled in phase space by the collaboration in the Wigner-Weyl approach. Together with Guy David, David Jerison and Hugo Duminil-Copin, she will explore the properties of the level sets of waves and of the localization landscape in disordered

• Marcel Filoche, Svitlana Mayboroda and Hugo Duminil-Copin will investigate the percolation properties of the level sets of the localization landscape, and especially their relation with the prediction of the mobility

• In cold atom experiments, Alain Aspect’s team, in collaboration with Marcel Filoche and Svitlana Mayboroda, will determine the mobility edge in 3D speckle potentials using their recently developed cutting-edge experimental platform, as well as the critical exponents of the Anderson

• In collaboration with James Speck, Doug Arnold will investigate the detailed structure of electronic eigenstates in disordered nitride-based alloys especially during the transition from localization to delocalization where they are expected to go through a multifractal nature at the mobility

• Claude Weisbuch and James Speck, in collaboration with Marcel Filoche, will experimentally assess electronic transport in nitride-based quantum Marcel Filoche’s team will develop a Wigner-Weyl approach to transport in the localization landscape theory, the last building block before achieving a quantum drift-diffusion model of semiconductors which promises to be the future design tool for semiconductor devices at the nanoscale.

• Richard Friend’s team, in collaboration with Marcel Filoche, will implement dynamical aspects of localization in the localization landscape theory and, in particular, investigate the role played by temperature in the absorption tails of organic compounds and in

We continue to be grateful to the Simons Foundation for this collaboration, which has provided us a forum to delve into these new developments. We look forward to sharing even more results of our efforts at the next Simons Foundation annual meeting in 2023.

Svitlana Mayboroda
University of Minnesota

Localization of Waves: Director’s Overview
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This talk is an overview of the progress and future plans of the Simons Collaboration on Localization of Waves.
 

James S. Speck
University of California, Santa Barbara

Unraveling the Inner Working of Gan-Based Light-Emitting Diodes
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Solid state lighting has been a disruptive technology for artificial light sources due to the remarkable efficiency, compact size, wide functionality, and accessibility. The technology is based on nitride-based semiconductors, particularly GaN and the alloy InxGa1-xN. In comparison to much more established group IV semiconductors (Si, Ge) and compound semiconductors (e.g., GaAs), the nitrides offer new physics due to their polar wurtzite crystal structure. InxGa1-xN alloys show strong natural compositional disorder that cannot be described the virtual crystal approximation. The alloy disorder directly impacts electron and hole transport and carrier recombination. In this talk, Speck will describe the emerging story of the internal physical processes in nitride alloys and devices, and the essential role of alloy disorder in the behavior of nitride alloys and nitride devices.
 

William D. Phillips
University of Maryland

The Coldest Stuff in the Universe: From Quantum Clocks to Quantum Simulators
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Cold atom (and cold molecule) physics has given us a new experimental platform with a variety of applications. Nanokelvin temperatures are now routine and are many orders of magnitude colder than any naturally occurring temperatures. Already, cold atoms have revolutionized atomic clocks, providing the basis for international timekeeping and providing unprecedented tests of fundamental theories. Cold atoms also provide a quantum system that is more easily controlled and measured than, for example, electrons in solids, offering the opportunity to better understand these complex systems. Some unsolved problems of many-body physics are being illuminated by these methods, as are phenomena such as Anderson localization.
 

Guy David
Université Paris-Saclay

Estimates for the Number of Eigenvalues of a Schrödinger Operator
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The landscape function (a solution of Lu=1 for the operator L at hand) has been used with success to get valuable information on the eigenfunctions, their localization and eigenvalues, for instance of Schrödinger operators. In this talk, David will describe different mathematical results that use that function and try to explain some of the features of the proofs and challenges for further estimates.
 

Richard Friend
University of Cambridge

Static and Dynamic Disorder in Organic and Perovskite Semiconductors
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Organic, molecular semiconductors are now successfully used in semiconductor devices such as LEDs for OLED displays and solar cells. Electronic overlap between adjacent molecules is relatively poor, so that disorder can readily cause spatial localization of electronic states and thus limit transport. Friend and collaborators have been exploring the different roles of static disorder and of dynamic disorder due to thermal populations of phonons. Where dynamic disorder is dominant, coupling of vibrational excitations to electronic excitations can allow novel regimes for high mobility transport, and they have investigated this in molecular systems that show long range exciton diffusion. Separately, they have been exploring static and dynamic disorder in the family of lead halide perovskites that are now used in efficient thin-film solar cells. These materials show unexpectedly clean semiconductor behavior despite high levels of structural disorder. Friend and collaborators have modeled static disorder in bromide/iodide alloy systems and have used the landscape formalism to explore compositional disorder on a wide range of length scales.
 

Camillo De Lellis
Institute for Advanced Study

The Mumford–Shah Functional and Its Challenges

The Mumford–Shah functional poses a very elegant and simple variational problem, which combines two of the most studied energies in the calculus of variations, the area functional and the Dirichlet energy. In spite of its simplicity, the regularity of minima is still a widely open field. De Lellis will review what is known about the simplest situation of two-dimensional minimizers, where a conjectural list of all possible local pictures is well-known and a complete epsilon-regularity theory is available.
 

Marcel Filoche
École Polytechnique

The Wigner–Weyl Approach to Wave Localization
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One of the most recent discoveries of the localization landscape theory relates to the structure of the landscape in phase space. We will present a new approach to localization combining the landscape approach with the Wigner–Weyl formalism, based on a new decomposition of phase space adapted to localized features. This approach allows us to predict extremely efficiently all energy-dependent quantities. It will be exemplified in two examples. In systems of cold atoms, it yields the first analytic prediction of spectral functions across the classical and quantum regime without adjustable parameters. In inorganic semiconductors, the incorporation of the landscape scheme in a drift-diffusion computation enables to speed up by a factor of 1,000 the computation of light absorption for three-dimensional random InGaN alloys in samples of side length 100nm, a computation otherwise beyond the reach of usual quantum computations. In this talk, Filoche will show how this approach will allow us in the future to investigate the delicate formation of excitons (electron-hole pairs) and bi-excitons, and how it will be tested experimentally in cold atom platforms.
 

Hugo Duminil-Copin
Université de Genève

A Perculation Perspective on Random Waves
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In this talk, Duminil-Copin will review recent progress in the understanding of nodal lines of random waves and related models. In particular, Duminil-Copin will discuss the possible emergence of giant nodal lines with complicated topological properties, thus partially answering a conjecture of Peter Sarnak. He will also discuss future research directions and possible connections between percolation theory and waves’ localization.