Laboratoire de Physique Théorique

de la Matière Condensée

Riccardo Ben Ali Zinati (LPTMC)

A bird's eye view on discrete universalities

(Séminaire zoom:   ID de réunion : 950 1508 7598 Code secret : 599922)

Our modern understanding of universality, namely the independence of the critical properties of a system from its microscopic details, is based on the ideas put forward by K. Wilson and formalised within the framework of the renormalization group (RG). Despite the centrality of the subject in modern days theoretical Physics and the many decades passed since Wilson's original works, the classification of universality classes is to date largely unsolved. In recent years, however, the functional reformulation of the perturbative RG provided a simple tool to approach such a classification, extending significantly our knowledge of universality classes entailed by discrete symmetry groups in scalar field theories. In this talk, I will first introduce the technique, elucidate its key-point features and explain why it is so effective in charting the theory space. In passing, I briefly review the theory of invariant polynomials paying particular attention to the so-called Hilbert (or Molien) series. I will then focus on three particular cases: the scalar field theories endowed with the symmetry group of regular polytopes, the randomly diluted Ising models and the more general hyper-cubic scalar field theories. For each of these three cases, I review the main results and suggest possible lines of future investigations.