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Abstract |
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.
References:
https://arxiv.org/abs/1902.05328
https://arxiv.org/abs/2006.12808 |
