Is the phase transition in the Heisenberg model described by the (2+ϵ)(2+\epsilon)-expansion of the nonlinear σ\sigma-model?

Nonlinear $\sigma$-model is an ubiquitous model. In this paper, the $O(N)$ model where the $N$-component spin is a unit vector, ${\bf S}^2=1$,is considered. The stability of this model with respect to gradient operators $(\partial_{\mu}{\bf S}\cdot \partial_{\nu}{\bf S})^s$, where the degree $s$ is arbitrary, is discussed. Explicit two-loop calculations within the scheme of $\epsilon$-expansion, where $\epsilon=(d-2)$, leads to the surprising result that these operators are relevant. In fact, the relevancy increases with the degree $s$. We argue that this phenomenon in the $O(N)$-model actually reflects the failure of the perturbative analysis, that is, the $(2+\epsilon)$-expansion. It is likely that it is necessary to take into account non-perturbative effects if one wants to describe the phase transition of the Heisenberg model within the context of the non-linear $\sigma$-model. Thus, uncritical use of the $(2+\epsilon)$-expansion may be misleading, especially for those cases for which there are not many independent checks.Comment: RevTex, 33 pages, figures embedde

Random Matrix Theories in Quantum Physics: Common Concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report

Disparités doctorales entre Nord et Sud : bilan et stratégies de « coordination juste » à l’UMI SOURCE (2022-2023)

Between 2022 and 2023, UMI SOURCE underwent a significant transformation in doctoral integration, aimed at enhancing cooperation and equity between its Northern and Southern branches. This document provides an assessment of these dynamics, highlighting the initial challenges, adopted strategies, and achieved successes. Initiatives such as monthly inter-branch workshops, the adoption of collaborative platforms, and the establishment of the Working Group on Doctoral Disparities and Inequalities (GTDID) have contributed to better integrating doctoral candidates and promoting collaborative and equitable research. Although there is still a path to tread, these actions have helped to overcome some initial reluctances related to cultural and geographical diversity, paving the way for new collaborations and joint research endeavors.Entre 2022 et 2023, l’UMI SOURCE a entrepris une transformation significative dans l’intégration doctorale, visant à améliorer la coopération et l’équité entre ses antennes du Nord et des Suds. Ce document présente un bilan de ces dynamiques, mettant en lumière les défis initiaux, les stratégies adoptées, et les succès obtenus. Les initiatives telles que les ateliers mensuels inter-antennes, l’appropriation de plateformes d’échanges, et la création du Groupe de Travail sur les Disparités et les Inégalités Doctorales (GTDID) ont contribué à proposer une meilleure intégration des doctorants et à promouvoir une recherche collaborative et équitable. Même si du chemin reste à parcourir, ces actions ont permis de surmonter quelques réticences initiales liées à la diversité culturelle et géographique, ouvrant la voie à de nouvelles collaborations et recherches communes