Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).
The critical-point exponent $\ensuremath{\eta}$ for a critical point of order
O in dimensions less than ${d}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, is calculated to leading nonvanishing order in the parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}{d}_{\mathcal{O}}\ensuremath{-}d$. The result is given for
n-component isotropically interacting magnetic systems. For Ising systems,
n=1, the result is ${\ensuremath{\eta}}_{\mathcal{O}}=\frac{{\ensuremath{\epsilon}}_{\mathcal{O}}^{2}4{(\mathcal{O}\ensuremath{-}1)}^{2}}{{({2\mathcal{O}}{\mathcal{O}})}^{3}}$. As
O increases, the coefficient of ${\ensuremath{\epsilon}}_{\mathcal{O}}^{2}$ rapidly becomes very small, varying as ${2}^{\ensuremath{-}6\mathcal{O}}$ for
O large. In the limit of large
n, ${\ensuremath{\eta}}_{\mathcal{O}}$ for odd order points approaches a constant and, for even order points, is proportional to
n1.
Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.