In 1958, the physicist P. W. Anderson predicted that random impurities could transform a conductive medium into an insulator, so that its electrons would become localized. This idea generated extremely rich, diverse, and important fields of research in physics, chemistry, and mathematics. In particular, the field of spectral analysis of disordered systems has been highly prolific in recent decades, both in terms of rigorous mathematical results with clear physical interpretations and in the number of theoretical and intrinsically interesting tools created to obtain such results.
The main topic of this workshop is concerned with spectral theory problems for ergodic Schrödinger type operators, which are used to mathematically model disordered systems. Besides the talks, it will feature an informal open problems session. The workshop also serves as a prologue to the thematic session on the Spectral Theory of Ergodic Schrödinger Operators, which is part of the 35th Brazilian Colloquium of Mathematics (35º Colóquio Brasileiro de Matemática).
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Abel Klein
Title:
Localization phenomena in the random XXZ spin chain: Finite chains
Abstract:
We study localization phenomena in the finite Heisenberg XXZ spin chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for the resolvent of the random many-body XXZ Hamiltonian. This property implies slow propagation of information for finite chains in the same energy interval. In the sequel of this talk in the Brazilian Colloquium of Mathematics we will discuss how this quasi-locality property implies localization in the infinite random XXZ spin chain. (Joint work with Alexander Elgart.)
Ao Cai
Title:
Lyapunov exponents of mixed random-quasiperiodic cocycles
Abstract:
In a joint project with Pedro Duarte (ULisboa) and Silvius Klein (PUC-RIO), we introduced and studied the model of mixed random-quasiperiodic cocycles motivated by a question of Jiangong You (CIM, NKU) in 2018 on the stability of the Lyapunov exponents of quasiperiodic Schrõdinger operators under random perturbations. In this talk, we will discuss some recent progress on it.
Constanza Rojas-Molina
Title:
Transition in the Integrated Density of States of a correlated random Schrödinger operator.
Abstract:
In 2017, Sabot, Tarrés and Zeng proved a connection between a reinforced random walk with a non-linear sigma-model coming from statistical mechanics, studied by Disertori, Spencer and Zirnbauer. In both models there is the presence of a random Schrödinger operator. We study the Integrated Density of States for this model and show it undergoes a transition depending on the dimension and the strength of the disorder, that is linked to the strength of the reinforcement parameter in random walks. This behavior is in stark contrast with the usual behavior of the IDS in disordered systems, known as Lifshitz tails, which are usually associated with Anderson Localization and pure point spectrum in the random operator. This is joint work with M. Disertori, X. Zeng, and V. Rapenne (Journal of Spectral Theory, 2025).