Scaling and the Renormalization Group

To go beyond Landau theory, the order parameter fluctuations need to be included. This can be achieved by writing the partition function as a functional integral 

we have also included an external field h conjugate to the order parameter (in the case of the ferromagnetic transition, / is a uniform magnetic field). The thermodynamic average m of the order parameter is given by the average (cj)) of the field with respect to the statistical weight 

In the disordered phase, the thermodynamic average of the order parameter vanishes, but its fluctuations are nonzero. When the critical point is approached, the spatial correlations of the order parameter fluctuations, as characterized by the correlation function G(x — y) = ((j)(x)())(y)), become long-ranged. Close to the critical point, their typical length scale, the correlation length diverges as 

In addition to the critical exponents v and yt, defined above, other exponents describe the dependence of the order parameter and its correlations on the distance from the critical point and on the field conjugate to the order parameter. The definitions of the most commonly used critical exponents are summarized in Table 1. These exponents are not all independent from each other. The four thermodynamic exponents a, p, y, 8 all derive from the 

The correlation length and correlation function exponents are related by 

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