Universality in the Renormalization Group

Let us now see how the formalism of the RG can explain universality.

Suppose we start from a system with a Hamiltonian ${\mathcal {H}}$${\mathcal {H}}$ which depends on some coupling constants $[K]$$[K]$; suppose also that we can write a RG transformation which in general gives rise to no more than $D$$D$ couplings. Under the action of the RG the initial physical Hamiltonian will move in the $D$$D$-dimensional theory space and follow the RG flow. Let us call ${\mathcal {H}}^{*}$${\mathcal {H}}^{*}$ the fixed point towards which ${\mathcal {H}}$${\mathcal {H}}$ tends, and assume it has only one relevant direction and $D-1$$D-1$ irrelevant ones; linearising the flow near ${\mathcal {H}}^{*}$${\mathcal {H}}^{*}$ we therefore identify $D-1$$D-1$ linearly independent vectors which constitute the hyperplane tangent in ${\mathcal {H}}^{*}$${\mathcal {H}}^{*}$ to its basin of attraction (i.e. the linearisation of the critical manifold near ${\mathcal {H}}^{*}$${\mathcal {H}}^{*}$). In general, if we "zoom out" and "look" also in regions far from the fixed point ${\mathcal {H}}^{*}$${\mathcal {H}}^{*}$, in theory space there will be a $D-1$$D-1$-dimensional critical manifold ${\mathcal {C}}$${\mathcal {C}}$. Let us now consider a generic model in theory space; remembering that in general $\beta =1/k_{B}T$$\beta =1/k_{B}T$ is included in the definition of the coupling constants $K_{i}$$K_{i}$, if we change the temperature of the system all the $K_{i}$$K_{i}$-s will change and thus the model will describe a trajectory ${\mathcal {P}}$${\mathcal {P}}$ in the $D$$D$-dimensional theory space, called physical subspace (because we can move along it by changing a physically accessible parameter like the temperature).

To make an explicit example, let us consider an Ising model with nearest- and next-nearest-neighbour interactions in the absence of any external field, so that:

$$-\beta {\mathcal {H}}=K_{1}\sum _{\left\langle i,j\right\rangle }S_{i}S_{j}+K_{2}\sum _{i,j={\text{n.n.n.}}}S_{i}S_{j}$$

$$-\beta {\mathcal {H}}=K_{1}\sum _{\left\langle i,j\right\rangle }S_{i}S_{j}+K_{2}\sum _{i,j={\text{n.n.n.}}}S_{i}S_{j}$$

In this case the physical subspace can be a straight line once the ratio $K_{1}/K_{2}$$K_{1}/K_{2}$ has been fixed, and this line goes from $K_{1}=K_{2}=0$$K_{1}=K_{2}=0$ (when $T=\infty$$T=\infty$) to $K_{1}=K_{2}=\infty$$K_{1}=K_{2}=\infty$ (when $T=0$$T=0$). The physical subspace will in general intersect the critical manifold ${\mathcal {C}}$${\mathcal {C}}$ in some point $[K_{c}]$$[K_{c}]$: this is the critical point of our system, and as it must be we have reached it by only varying its temperature.

Now, if we consider two different physical systems they will be characterized by two different physical subspaces ${\mathcal {P}}_{1}$${\mathcal {P}}_{1}$ and ${\mathcal {P}}_{2}$${\mathcal {P}}_{2}$ in theory space. In general, they will intersect the critical manifold ${\mathcal {C}}$${\mathcal {C}}$ in different points, and so they will have different critical temperatures; however, under the action of the RG they will flow towards the same critical fixed point, and since (as we are going to show explicitly in the following section) the critical behaviour of a system is determined by the properties of the RG flow near a fixed critical point, these two systems will behave identically near their critical points. This is how universality is explained within the RG.

In the case of the Ising model we are considering, the situation can be represented in the two-dimensional theory space as follows:

RG flow for the considered Ising model