Critical behavior of the ising model close to four dimensions

Abstract

In this chapter, we shall use the momentum space RG with a sharp cutoff to study the critical behavior of the Ising model close to four dimensions, where the parameter ε = 4- D is small. In this case the critical exponents and other physical quantities can be expanded in a series in powers of ε (Wilson et al. 1972). For sufficiently small ε, a truncation of this expansion retaining only a small number of terms yields quantitatively accurate results. Of course, in order to calculate the critical exponents in the physically relevant dimension D = 3, one has to calculate as many terms in the ε-expansion as possible and then extrapolate the series to ε = 1 (Guida and Zinn-Justin 1998, Zinn-Justin 2002, Pelissetto and Vicari 2002). In order to explain this method, we shall focus here on the calculation of the correlation length exponent v to linear order in ε. To this end, it is convenient to carry out the mode-elimination step in momentum space, using a sharp momentum-shell cutoff. The relevant integrals will be calculated perturbatively in powers of the interaction. In order to keep track of the various terms generated in perturbation theory, it is useful to represent them in terms of Feynman diagrams. We therefore begin this chapter with a brief but self-contained introduction to diagrammatic perturbation theory. © 2010 Springer-Verlag Berlin Heidelberg.