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arXiv:2609.39462

Many-body localization: an introduction for mathematicians

François Huveneers

Published Sep 30, 2026• 0 citations
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Abstract

These notes are based on a mini-course delivered at the Les Houches Summer School, Quantum Theory on All Scales, in August 2026. They introduce many-body localization to mathematicians, emphasizing the physical mechanisms that support or destabilize it. The mathematical discussion deals with how perturbative rotations, renormalization, and probabilistic estimates are used in a rigorous approach to many-body localization.

DOI
10.48550/arxiv.2609.39462

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These notes are based on a mini-course delivered at the Les Houches Summer School, Quantum Theory on All Scales, in August 2026. They introduce many-body localization to mathematicians, emphasizing the physical mechanisms that support or destabilize it. The mathematical discussion deals with how perturbative rotations, renormalization, and probabilistic estimates are used in a rigorous approach to many-body localization.
These notes accompany lectures given in June 2025 at the summer school \emph{Fundamental Problems in Statistical Physics XVI}. They offer a short introduction to random quantum circuits as simple models for generic many-body quantum systems. They give an outline of the motivation for introducing these models, starting from ideas of random matrix theory. They also provide a sketch of calculations of some of the quantities of most physical interest, based on an average over an ensemble of systems. These quantities give insights into operator spreading, entanglement dynamics and spectral correlations.
These are the notes for the 4.5-hour course with the same title that I delivered in August 2025 at the Les Houches summer school ``Exact Solvability and Quantum Information''. In these notes I pedagogically introduce the setting of brickwork quantum circuits and show that it provides a useful framework to study non-equilibrium quantum many-body dynamics in the presence of local interactions. I first show that brickwork quantum circuits evolve quantum correlations in a way that is fundamentally similar to local Hamiltonians, and then present examples of brickwork quantum circuits where, surprisingly, one can compute exactly several relevant dynamical and spectral properties in the presence of non-trivial interactions.