Critical Behaviour from the Field Theoretical Renormalization Group Techniques

Victor A. Alessandrini, H. Devega, F Schaposnik

C. Z. Yang, Z. Y. Li

Abstract It is shown that the mean field renormalization group techniques provide a reasonable description of the critical behaviour of the site‐bond diluted Ising model, both in the two‐dimensional system and the surface of a semi‐infinite three‐dimensional system.

H. Wagner

Jean Zinn‐Justin

The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].

Kenneth Geddes Wilson