Vortex – Spin systems with discrete and continuous symmetry: topological defects, Bayesian statistics, quenched disorder and random fields

Le projet Vortex a l’ambition de comprendre la géométrie aléatoire sous-jacente aux "transitions de phase topologiques" qui ont été découvertes dans les années 1970 par Berezinskii, Kosterlitz et Thouless.

Vortex est un projet en mathématiques qui se situe au carrefour des probabilités et de la physique mathématique. L'un des buts principaux de ce projet est de comprendre la géométrie aléatoire sous-jacente aux "transitions de phase topologiques" qui apparaissent dans des "systèmes de spins" simples à définir mais dont les propriétés sont étonnamment riches.
Christophe Garban étudiera la géométrie fractale aléatoire qui apparaît au cours de ces transitions de phase en développant de nouveaux outils et en créant des liens avec les statistiques bayésiennes.

Key information / Informations

•    Funding programme / Programme de financement: Horizon Europe - ERC Consolidator Grant - 2021
•   Coordinator / Coordinateur: Université Claude Bernard Lyon 1 (UCBL), Pr Christophe GARBAN (ICJ)
•   Partners’ list / Liste des partenaires :

o Centre National de la Recherche Scientifique (CNRS)

o Universidad de Chile (UCHILE)

•   Budget : 1 616 250 €
•   Grant / Subvention :1 616 250 €
•   Début – Fin : 2023-2028

Context / Contexte

The main goal of this project is to understand the geometry of the deeply influential topological phase transitions which were discovered in the 70's by Berezinskii, Kosterlitz and Thouless. The archetypal example of such phase transitions arises in the 2d XY model in which topological defects, called vortices, behave very differently at small and high temperature. The mathematical understanding of this rich phenomenon goes back to the work of Fröhlich and Spencer in the 80's and involves the 2d Coulomb gas. This project is aimed at analyzing this phase transition through the prism of random fractal geometry by associating natural percolating sets to the XY model whose behavior will depend crucially on the temperature. One constant source of inspiration will be the deep geometric content and powerful probabilistic methods gathered over the last 20 years for celebrated discrete symmetry models such as 2d critical Ising or percolation. Vortex will bring new tools and make connections with Bayesian statistics.

Objectives / Objectifs

The main goal of this project is to understand the geometry of the deeply influential topological phase transitions in i) bringing new tools such as the cable XY model ii) identifying new interesting phase transitions which arise in discrete symmetry spin systems due to the presence of topological defects iii) developing a very promising connection with works on Bayesian statistics and iiii) investigating the mechanisms at the crossroads of Quantum Field Theory, noise sensitivity and scaling limits of random growth models.

Expected impact and results / Impact et résultats attendus

Vortex will explore the geometry of the deeply topological phase transitions by making several novel and fruitful connections with the dimer and Ising models. The new connections made with statistical reconstruction and Bayesian statistics will give access to the even more fascinating and least understood world of spin systems with non-Abelian (gauge-symmetry).
Finally, the project will investigate the mechanisms which relate the microscopic background noise with the large-scale structures it induces in the contexts of Quantum Field Theory and KPZ fixed point. The impact of this project will go well beyond the current understanding of topological phase transitions in a wide variety of settings where they arise.

LIP’s contribution / Rôle de LIP

LIP a accompagné le porteur de projet dans toute la phase de montage, coordination du montage administratif et financier du projet, conseil à la rédaction. LIP accompagne l’Université Lyon 1 pour le suivi administratif et financier du projet.