Understanding universality from the linearized RG theory

In the linearized RG theory, at a fixed point, all scaling variables are 0, whether relevant, irrelevant or marginal. Consider the case where there are no marginal scaling variables. Recall, moreover, that irrelevant scaling variables will iterate to 0 under repeated RG transformations, starting from a point near the unstable fixed point, while the relevant variables will be driven away from 0. These facts provide us with a formal procedure for locating the fixed point:

i.

Start with the space spanned by the full set of eigenvectors of T.

ii.

Project out the relevant subspace by setting all the relevant scaling variables to 0 by hand.

iii.

The remaining subspace spanned by the irrelevant eigenvectors of T defines a hypersurface in the full coupling constant space. This is called the critical hypersurface.

iv.

Any point on the critical hypersurface belongs to the irrelevant subspace and will iterate to 0 under successive RG transformations. This will define a trajectory on the hypersurface that leads to the fixed point as illustrated below:

  
Figure 4:

This fixed point is called the critical fixed point. Note that it is stable with respect to irrelevant scaling variables and unstable with respect to relevant scaling variables.

What is the importance of the critical fixed point? Consider a simple model in which there is one relevant and one irrelevant scaling variable, and , with corresponding couplings and . In an Ising-type model, might represent the reduced nearest neighbor coupling and might represent a next nearest neighbor coupling. Relevant variables also include experimentally tunable parameters such as temperature and magnetic field. The reason is relevant and is irrelevant is that there must be at least nearest neighbor coupling for the existence of a critical point and ordered phase at h=0 but this can happen whether or not there is a next nearest neighbor coupling. Thus, the condition

defines the critical surface, in this case, a one-dimensional curve in the - plane as illustrated below:

  
Figure 5:

Here, the blue curve represents the critical ``surface'' (curve), and the point where the arrows meet is the critical fixed point. The full coupling constant space represents the space of all physical systems containing nearest neighbor and next nearest neighbor couplings. If we wish to consider the subset of systems with no next nearest neighbor coupling, i.e., , the point at which the line intersects the critical surface (curve) defines the critical value, and corresponding critical temperature and will be an unstable fixed point of an RG transformation with . Similarly, if we consider a model for which , but having a fixed finite value, then the point at which this line intersects the critical surface (curve) will give the critical value of for that model. For any of these models, lies on the critical surface and will, under the full RG transformation iterate toward the critical fixed point. This, then, defines a universality class: All models characterized by the same critical fixed point belong to the same universality class and will share the same critical properties. This need not include only magnetic systems. Indeed, a fluid system, near its critical point can be characterized by a so called lattice gas model. This model is similar to the Ising model, except that the spin variables are replaced by site occupance variables , which can take on values 0 or 1, depending on whether a given lattice site is occupied by a particle or not. The grand canonical partition function is

and hence belongs to the same universality class as the Ising model. The critical surface, then, contains all physical models that share the same universality properties and have the same critical fixed point.

In order to see how the relevant scaling variables lead to the scaling relations among the critical exponents, we next need to introduce the scaling hypothesis.