Wilson-Fisher fixed point

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Author: Prof. Jean Zinn-Justin, CEA, IRFU and Institut de Physique Théorique, Centre de Saclay, F-91191 Gif-sur-Yvette, France

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Prof. Jean Zinn-Justin accepted the invitation on 23 October 2008 (self-imposed deadline: 23 February 2009).

Wilson-Fisher fixed point refers to the theory of phase transitions and critical phenomena and the study of their universal properties using renormalization group arguments. Large distance behaviour is then related to fixed points of the renormalization group. Wilson and Fisher managed to determine the fixed points relevant for a large classs of phase transitions (liquid-vapour, Helium, ferromagnets...) by using a method that allows, in quantum field theory, to define perturbation theory in the form of Feynman diagrams, for complex values of the space dimension $d$ . They found that near dimension four ( $d=4-\varepsilon$ ), universal quantities could be calculated in the form of an $\varepsilon$ -expansion ( $\varepsilon >0$ ). This provided the first example of an analytic calculation of critical exponents that differed from their classical or mean-field or quasi-Gaussian values.