Introduction to Renormalization Group (RG) Calculations

In our last lesson, we built the conceptual framework of the Renormalization Group, exploring the core ideas of coarse-graining, scale invariance, and fixed points. We saw that the RG provides a powerful language for describing how physical laws transform across different scales. Now, we will move from concept to practice.

This lesson is dedicated to making these ideas concrete by performing explicit RG calculations. Our objective is to apply the coarse-graining procedure to two different systems, demonstrating the two main flavors of the technique. First, we will tackle a real-space RG calculation for the 1D Ising model, a classic example from statistical mechanics. Then, we will perform a momentum-shell RG calculation for a scalar field theory, which will connect directly to your studies in Quantum Field Theory. By the end, you will have seen how the abstract idea of "flow" in the space of Hamiltonians emerges from a concrete mathematical procedure.

Real-Space RG: Decimation in the 1D Ising Model

The simplest setting to see the RG in action is a one-dimensional lattice model. Here, the "coarse-graining" procedure is particularly intuitive. We will use a method called decimation, which involves systematically removing a fraction of the degrees of freedom (the spins) and seeing how the interactions between the remaining ones are modified.

Consider the 1D Ising model, whose Hamiltonian you've seen before:

H = - J i \sum s_{i} s_{i + 1}

where $s_{i} = \pm 1$ . The probability of a configuration ${s_{i}}$ is given by the Gibbs measure, $p ({s_{i}}) \propto e^{- β H} = e^{K \sum_{i} s_{i} s_{i + 1}}$ , with $K = β J$ .

Our decimation procedure will be to "trace out" (i.e., sum over) all the spins at odd-numbered sites, leaving us with a new system composed only of the spins at even-numbered sites. This new system will look like a 1D Ising model with a larger lattice spacing, and our goal is to find its new effective coupling, $K^{'}$ .

A visual representation of coarse-graining a 2D Ising model configuration at its critical temperature. The original 64x64 lattice (left) is successively blocked into 32x32 and 16x16 lattices. Note the statistical self-similarity: the pattern of spin domains looks qualitatively the same at each scale. Our 1D calculation is a simpler version of this process.

The following reading provides a clear, step-by-step derivation of the RG transformation for this model.

Real-Space Renormalization Group Explained - About

This blog post by E. Regis clearly walks through the real-space RG calculation for the 1D Ising model. It's an excellent example of turning the conceptual steps into a concrete calculation.

Please read the following sections: Decimating The 1-D Ising Model: This sets up the problem, defining the Hamiltonian and the decimation procedure. Finding the renormalized measure: Follow the mathematical steps here carefully. The key idea is to sum over the odd-sited spins (s_{2I+1}) that are "sandwiched" between two even-sited spins (s'_{I} and s'_{I+1}). This sum defines the new effective interaction, \psi(s'_I, s'_{I+1}). Solving the inverse problem: This is the final step. We demand that the new probability distribution has the same functional form as the original, which allows us to solve for the new coupling K' in terms of the old one, K.

The central result of this calculation is the recursion relation, or RG flow equation:

K^{'} = \frac{1}{2} ln (cosh (2 K))

This equation tells us exactly how the coupling constant transforms under one step of our coarse-graining procedure. If we start with a system with coupling $K$ , after "zooming out" by a factor of two, it looks like a system with coupling $K^{'}$ . We have generated an RG flow. We will analyze the structure of this flow in the next lesson.

Momentum-Shell RG: The Wilsonian Approach in Field Theory

Now let's translate this procedure into the language of field theory, which should feel familiar from your QFT course. Here, instead of a lattice of spins, our degrees of freedom are the Fourier modes $ϕ (k)$ of a field. Short-distance physics corresponds to high-momentum ("fast") modes, while long-distance physics is described by low-momentum ("slow") modes.

Coarse-graining in this context, as pioneered by Kenneth Wilson, means integrating out the field modes in a high-momentum "shell" to find an effective action for the remaining low-momentum modes.

We'll consider a real scalar field theory in Euclidean space with a $ϕ^{4}$ interaction, our standard workhorse model:

S [ϕ] = \int d^{d} x (\frac{1}{2} (\nabla ϕ)^{2} + \frac{1}{2} m_{0}^{2} ϕ^{2} + \frac{λ _{0}}{4 !} ϕ^{4})

We assume this theory is valid up to some physical momentum cutoff $Λ$ . The RG procedure consists of three steps:

Decomposition: Split the field $ϕ (k)$ into a low-momentum part ( $ϕ_{<}$ ) for $∣ k ∣ < b Λ$ and a high-momentum part ( $ϕ_{>}$ ) for $b Λ < ∣ k ∣ < Λ$ , where $b < 1$ .
Integration: Calculate a new effective action for $ϕ_{<}$ by performing the path integral over the high-momentum modes $ϕ_{>}$ : $e^{- S_{e f f} [ϕ_{<}]} = \int D ϕ_{>} e^{- S [ϕ_{<} + ϕ_{>}]}$
Rescaling: Rescale momentum and fields to make the new action look like the old one, allowing for direct comparison. (We will focus on step 2 for now).

The following video provides a detailed walkthrough of this calculation. It is dense, but it is a perfect illustration of the "craft of theoretical physics" applied to the RG.

Wilsonian Renormalization - QFT II, Part 23

This lecture from the IFT-UNESP "QFT II" series provides a detailed, technical derivation of the Wilsonian RG for \phi^4 theory. It demonstrates how integrating out high-momentum modes modifies the parameters of the low-energy theory.

The calculation is involved, so we will break it down into key stages. First, watch the formal setup. This part introduces the core idea of momentum-shell integration. The lecturer splits the field \phi into a low-energy part (which he also calls \phi) and a high-energy part (\phi_{\text{hat}}). He then rewrites the generating functional Z to separate the path integral over these two sets of modes. Next, focus on the actual calculation of how the low-energy effective action is modified. The lecturer shows this using Feynman diagrams, where internal loops involving the high-energy field \hat{\phi} generate new effective interactions for the low-energy field \phi. Watch the segment on the mass renormalization from this calculation. Here, a one-loop "tadpole" diagram involving \hat{\phi} fields leads to a correction to the mass term for the \phi field. The original m_0^2 \phi^2 term becomes (m_0^2 + \delta m^2)\phi^2, where \delta m^2 comes from integrating out the high-momentum shell. Then, watch the segment on the coupling constant renormalization from this calculation. A one-loop "bubble" diagram involving \hat{\phi} fields generates a correction to the \phi^4 vertex. The original coupling \lambda_0 is modified to \lambda_0 + \delta\lambda.

This calculation, though perturbative, is the heart of the Wilsonian RG. It shows explicitly that the effect of short-distance physics (the $ϕ^$ loops) is to redefine the parameters (mass and coupling) of the long-distance effective theory. Unlike the infinities you might be used to in QFT, these corrections are finite because the loop integral is restricted to a finite momentum shell, from $b Λ$ to $Λ$ .

Conclusion

In this lesson, we have rolled up our sleeves and performed two fundamental RG calculations. Despite their different appearances, both procedures followed the same logic:

Separate the degrees of freedom into "slow" (long-wavelength) and "fast" (short-wavelength) parts.
Integrate out the fast modes.
Find the new effective theory for the slow modes, which will have renormalized couplings.

Key Takeaways:

Real-space RG on the 1D Ising model via decimation yields a discrete recursion relation, $K^{'} = f (K)$ , which explicitly defines the RG flow.
Momentum-shell RG on $ϕ^{4}$ theory shows how integrating out high-momentum field modes in a path integral generates corrections to the parameters (mass, coupling) of the low-energy effective action.
In both cases, we see that the process of coarse-graining modifies the fundamental constants of the theory. This transformation of the Hamiltonian/Lagrangian is the Renormalization Group flow.

We now have the tools to generate the RG flow. The next step, and the subject of our next lesson, is to analyze its structure. We will learn how to find the fixed points of the flow and analyze their stability, which is the key to understanding phase transitions and universality.

Recommended resources

Real-Space RG: Decimation in the 1D Ising Model

Momentum-Shell RG: The Wilsonian Approach in Field Theory

Conclusion