New directions in disordered systems: A conference in honor of Abel Klein

In July 2022, the conference New Directions in Disordered Systems: In honor of Abel Klein, took place at CY Cergy Paris Université Neuville Campus, to celebrate the contributions of Abel Klein to mathematical physics. The meeting was attended by colleagues, co-authors, former students and postdocs, and many admirers of his research, from around the world. Originally planned to take place in 2020, but postponed because of COVID-19, this was, for many, the first big conference after almost two years of isolation and remote work, adding to the excitement of celebrating Abel’s work. Abel Klein is a major figure in the theory of quantum disordered systems with a career spanning five decades (and counting). After studies in Rio de Janeiro, Brazil, he obtained his Ph.D. in mathematics under the direction of mathematical physicist Irving E. Segal at MIT in 1971. After stays at UCLA and Princeton University, he moved to the University of California, Irvine, where he has been since 1974.

Abel’s outstanding research spans constructive quantum field theory, statistical mechanics, and solid-state physics. His work has been incredibly influential in mathematical physics and to the people around him. Abel’s papers are sophisticated, technically versed, and carefully written in order to achieve a complete and comprehensible exposition.

Abel has been a lifelong supporter, as colleague and friend, to many young researchers in mathematics. He has successfully mentored many doctoral students and postdocs throughout his career, the list including a very high level research administrator, ICM speakers, academy members, editors of high level journals, Chaired Professors at their departments, and winners of Sloan fellowship and major prizes. He has also mentored many others informally: students and postdocs of others, young faculty, and young researchers from other places. He probably has not thought of most of it as mentorship: he is just someone very genuinely enjoying helping others, sharing his wisdom, and becoming friends. Yet mentorship it is, and quite a number of people became successful in the profession through empowerment by his sincere encouragement, by following his advice. And also by looking up to him.

During the conference New Directions in Disordered Systems, Abel’s colleagues, collaborators, mentees, formal and informal, and friends from around the world gathered to celebrate his career achievements with toasts and talks on their most recent works in the field, as can be seen in this Special Issue.

Abel is particularly known for his signature contributions to the theory of random Schrödinger operators (RSO), the research area he has worked on since its inception in the mid-1980s. His œvre consists of more than 120 papers, including publications in elite mathematical journals such as Annals of Mathematics, Inventiones Mathematicae, and the Journal of the European Mathematical Society, among others.

The theme of localization had already made appearances in some of Abel’s earliest works. Inspired by supersymmetric field theory and his work on the Parisi-Sourlas method, Abel and his collaborators tackled questions of localization in the Anderson model and regularity of the associated density of states. A technically involved but powerful approach to these questions in dimension one using Grassman variables has been implemented in Refs. 1 and 2. In localization in the ground-state of the one-dimensional X–Y model with a random transverse field,³ Abel and Perez proved exponential decay for the spin-spin correlation function using methods that re-appear in current studies on many-body localization for the XY spin chain.

One of the major open problems in mathematical physics is to prove or disprove the existence of the phase transition in 3D RSO, a conjecture universally accepted in the physics community and first posited by Anderson in his famous 1958 paper.⁴ In his tour de force work, “Extended states in the Anderson model on the Bethe lattice,”⁵ Abel was the first to establish a nonempty absolutely continuous component in the spectrum of a RSO. This was the first example of an ergodic operator having both pure point and absolutely continuous spectrum almost surely, but more importantly, it remains to this day essentially the only Anderson model where absolutely continuous spectrum has been proved. This work has inspired very fruitful research by other groups in the last three decades.

Abel returned to the topic of delocalization later as well, showing the existence of the phase transition in his landmark paper, “Dynamical delocalization in random Landau Hamiltonians,”⁶ co-authored with Germinet and Schenker. In this work, topological invariants associated with the transport properties in the integer quantum Hall effect, combined with detailed estimates of localization in the broadened Landau bands, were masterfully exploited to demonstrate the existence of dynamical mobility edges for the 2D Landau magnetic RSOs.

Abel is perhaps best known for his fundamental contributions to the development of the multiscale analysis (MSA) method and its application to localization for RSO. The work with von Dreifus and Klein⁷ is foundational for this method, as it created a mathematically convenient and accessible framework for the ideas of Fröhlich and Spencer.⁸ Pursuing this direction, Abel had an extremely productive collaboration with Germinet, co-authoring 24 research papers. Their paper, “Bootstrap multiscale analysis and localization in random media,”⁹ simplified and improved the MSA method. Abel and Germinet continued to develop these ideas, incorporating those in the work of Bourgain and Kenig,¹⁰ and Abel’s previous work¹¹ with Germinet and Hislop on localization for Poisson random potentials. Abel’s article with Germinet, “A comprehensive proof of localization for continuous Anderson models with singular random potentials,”¹² presents optimal results guaranteeing localization bounds with minimal assumptions on the probability distribution. This work¹² still presents the state-of-the-art for the MSA method.

A key ingredient in the proofs of localization with singular random potentials are the so-called unique continuation principles, as perfected in Ref. 10, that can be seen as concentration estimates for the eigenfunctions. These unique continuation principles have several applications in RSO, including the study of the density of states measure. Together with Bourgain, Abel established surprising and remarkably general results on the regularity of the density of states outer measure¹³ using such estimates, in particular, proving a long outstanding problem of continuity of the density of states in the continuum in dimensions two and three. Another of Abel’s ground-breaking results concerns the famous Mott ac-conductivity formula. With Lenoble and Muller, Abel published “On Mott’s formula for the ac-conductivity in the Anderson model.”¹⁴ The proof relies on a rigorous linear response theory for disordered systems, developed previously by Abel and co-authors.¹⁵ 

The bulk of Abel’s recent contributions have been concentrated in many-body localization (MBL), an area of intensive research in condensed matter physics. With Elgart and Stolz,^16,17 Abel proved dynamical localization, absence of information propagation, and exponential clustering in the droplet spectrum of the disordered XXZ spin chain, the prototypical system in the studies of MBL. His most recent contribution is an extension of some of these results to arbitrary (but scale independent) energies, in collaboration with Elgart.^18,19

To conclude, let us list some of the most outstanding challenges that remain unsolved in the theory of random Schrödinger operators today. These include proving/disproving the existence of a phase transition between localized and delocalized states in dimension three and above and establishing localization for the whole spectrum in dimension two. The related open challenge is to properly define and study many-body localization. Though these problems are very hard, they continue to attract ambitious researchers to the field, drive new research, and no doubt, will lead to new exciting discoveries.

Abel’s work and attitude toward research has inspired generations of mathematical physicists to face these challenging new problems. Quantum systems are as relevant as ever, with the fast-paced developments of quantum materials, like graphene, topological insulators, and quantum information theory. We expect that randomness, ubiquitous in our world, will continue to pose challenges that our field will tackle.

The authors would like to thank P. Müller for his contributions to this editorial. P.D.H. is partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 843327. C.R.-M. acknowledges ANR RAW ANR-20-CE40-0012-01 for financial support. The authors thank the institute CY Advanced Studies and their staff for their hospitality and support.