Momentum-Shell Renormalization Group for Landau-Ginzburg-Wilson Hamiltonian

Hello! Welcome to the next lesson in our exploration of the renormalization group.

In our last lesson, we accomplished a crucial task: we mapped the microscopic Ising model onto a continuous field theory, the Landau-Ginzburg-Wilson (LGW) ${\varphi }^4$ Hamiltonian. This gave us a universal starting point for analyzing critical phenomena, described by the functional integral:

$$Z=\int \mathcal{D}\varphi (\mathbf{x})\exp \left(-\int d^{d}x\left[\frac{r}{2}{\varphi }^2+u{\varphi }^4+\frac{c}{2}(\nabla \varphi {)}^2-h\varphi \right]\right)$$

This formulation, while powerful, is still difficult to solve due to the interacting ${\varphi }^4$ term. In this lesson, we will develop the systematic procedure for analyzing this theory, moving beyond the limitations of the real-space RG we discussed earlier.

Our learning outcome is to set up the momentum-shell RG procedure for the Landau-Ginzburg-Wilson Hamiltonian. This involves defining a precise, three-step algorithm that tells us how the effective physics changes as we integrate out short-distance fluctuations. This procedure forms the core of Ken Wilson's Nobel Prize-winning work.

1. The LGW Hamiltonian in Momentum Space

The RG procedure is most naturally performed in momentum space, where the gradient term becomes a simple quadratic in momentum, $\mathbf{q}$. You are well-acquainted with Fourier transforms from your work in QFT and condensed matter physics. Applying a Fourier transform to our field $\varphi (\mathbf{x})$, we can rewrite the LGW Hamiltonian (with $c=1$ for simplicity, as is conventional):

$$\mathcal{H}[\varphi ]=\frac{1}{2}{\int }^{\Lambda }\frac{d^{d}q}{(2\pi {)}^d}(r+q^2)|\varphi (\mathbf{q}){|}^2+u{\int }^{\Lambda }\frac{d^d{q}_1}{(2\pi {)}^d}\dots \frac{d^d{q}_4}{(2\pi {)}^d}\delta ({\mathbf{q}}_1+{\mathbf{q}}_2+{\mathbf{q}}_3+{\mathbf{q}}_4)\varphi ({\mathbf{q}}_1)\varphi ({\mathbf{q}}_2)\varphi ({\mathbf{q}}_3)\varphi ({\mathbf{q}}_4)$$

A few key points about this form:

• Momentum Cutoff $\Lambda$: The integrals are not over all momenta. There's an upper limit, or UV cutoff, $\Lambda$, which is a relic of the microscopic lattice spacing of the original model (e.g., $\Lambda \sim \pi /a$). It represents the smallest length scale we consider.
• Gaussian Part: The $(r+q^2)|\varphi (\mathbf{q}){|}^2$ term is the "free" or Gaussian part. Without the interaction, the modes $\varphi (\mathbf{q})$ at different momenta would be completely independent.
• Interaction Part: The $u{\varphi }^4$ term couples different momentum modes together, making the problem non-trivial. The delta function ensures momentum conservation in these interactions.

Our goal is to understand how the physics described by this Hamiltonian behaves at long distances, i.e., for small momenta $q\ll \Lambda$.

2. The Wilsonian RG Procedure

The central idea, developed by Wilson, is to systematically eliminate the high-momentum degrees of freedom and see what effect this has on the remaining low-momentum degrees of freedom. This is done in a three-step process.

The following image provides a helpful schematic of how coarse-graining in real space corresponds to integrating out high-momentum modes in momentum space.

The following reading provides a clear and concise description of the three steps we are about to detail.

Now, let's formalize these three steps.

Step 1: Coarse-Graining (Integrating out "fast" modes)

We begin by splitting our field $\varphi (\mathbf{q})$ into two parts. We choose a scaling factor $b>1$ and define a new, lower cutoff ${\Lambda }^{\prime }=\Lambda /b$.

• "Slow" modes ${\varphi }_{<}$: These are the low-momentum modes with $0\le |\mathbf{q}|<\Lambda /b$.
• "Fast" modes ${\varphi }_{>}$: These are the high-momentum modes with $\Lambda /b\le |\mathbf{q}|<\Lambda$.

Our field is simply the sum: $\varphi (\mathbf{q})={\varphi }_{<}(\mathbf{q})+{\varphi }_{>}(\mathbf{q})$. The partition function can be written by separating the functional integral over these two sets of modes:

$$Z=\int \mathcal{D}{\varphi }_{<}\int \mathcal{D}{\varphi }_{>}{e}^{-\mathcal{H}[{\varphi }_{<},{\varphi }_{>}]}$$

The goal of this first step is to perform the inner integral over the "fast" modes ${\varphi }_{>}$. This gives us a new, effective Hamiltonian ${\mathcal{H}}^{\prime }$ that depends only on the "slow" modes ${\varphi }_{<}$:

$$e^{-{\mathcal{H}}^{\prime }[{\varphi }_{<}]}=\int \mathcal{D}{\varphi }_{>}{e}^{-\mathcal{H}[{\varphi }_{<},{\varphi }_{>}]}$$

This integral is the most difficult part of the RG calculation. Because of the ${\varphi }^4$ interaction term, the fast and slow modes are coupled, so the integral over ${\varphi }_{>}$ will change the form of the Hamiltonian for ${\varphi }_{<}$. In the next lesson, we will see how to compute this integral perturbatively. For today, the key concept is that this integration step "renormalizes" the parameters of the theory. For instance, the original parameter $r$ will be transformed into a new parameter $r^{\prime }$ that incorporates the effects of the fast fluctuations.

Step 2: Rescaling Momenta

After Step 1, we have an effective theory for ${\varphi }_{<}$ that is defined up to a cutoff $\Lambda /b$. To compare this new theory to our original one (which had a cutoff $\Lambda$), we must put them on an equal footing. We do this by rescaling all momenta.

We define a new momentum variable ${\mathbf{q}}^{\prime }$ such that:

$${\mathbf{q}}^{\prime }=b\mathbf{q}$$

Now, as $\mathbf{q}$ ranges from $0$ to $\Lambda /b$, our new variable ${\mathbf{q}}^{\prime }$ ranges from $0$ to $\Lambda$. We have restored the original momentum cutoff. In real space, this momentum scaling corresponds to a length rescaling ${\mathbf{x}}^{\prime }=\mathbf{x}/b$. This is the "zooming out" part of the process, where we look at the system on a larger length scale.

Step 3: Rescaling the Field

The momentum rescaling in Step 2 alters the coefficients in the Hamiltonian. Specifically, the gradient term, which was $q^2|\varphi (\mathbf{q}){|}^2$, becomes $(q^{\prime }/b{)}^2|{\varphi }_{<}(q^{\prime }/b){|}^2$. To maintain a canonical form for the effective Hamiltonian, we perform a final rescaling of the field itself.

We define a new, rescaled field ${\varphi }^{\prime }({\mathbf{q}}^{\prime })$ as:

$${\varphi }^{\prime }({\mathbf{q}}^{\prime })={\zeta }^{-1}{\varphi }_{<}(\mathbf{q})={\zeta }^{-1}{\varphi }_{<}({\mathbf{q}}^{\prime }/b)$$

The rescaling factor $\zeta$ is chosen specifically to restore the coefficient of the gradient term ($q^{\prime 2}$) to its original value (e.g., 1). This ensures that the new Hamiltonian ${\mathcal{H}}^{\prime }[{\varphi }^{\prime }]$ has the same functional form as the original $\mathcal{H}[\varphi ]$, but with new, renormalized parameters $\{r^{\prime },u^{\prime },\dots \}$.

Test your understanding!

What is the primary motivation for each of the three steps in the momentum-shell RG procedure? Why can't we just integrate out the high-momentum modes (Step 1) and stop there?

Show answer

• Step 1 (Integrate): This is the core physical step. It removes the short-wavelength fluctuations and incorporates their effect into the parameters of the theory for the remaining long-wavelength modes.
• Step 2 (Rescale Momenta): This step is for comparison. It maps the momentum range of the new effective theory back to the original range, so that the new Hamiltonian can be directly compared to the old one. It's like resizing an image to its original dimensions to see what details have been lost or changed.
• Step 3 (Rescale Field): This step is for standardization. It brings the kinetic/gradient term of the Hamiltonian back into a canonical form, making the transformation of the other parameters (like mass and coupling constant) unambiguous.

We can't stop after Step 1 because the resulting theory describes physics at a different scale (cutoff $\Lambda /b$) and with a non-standard form. The rescaling steps are essential for defining a repeatable transformation whose iteration and fixed points can be studied.

3. The RG Transformation and Flow

Together, these three steps define a mapping or transformation, denoted ${\mathcal{R}}_b$, in the space of all possible Hamiltonians (or "theory space"). This transformation takes the parameters of our original theory, $\{r,u\}$, and maps them to a new set of parameters, $\{r^{\prime },u^{\prime }\}$, for the rescaled theory:

$$\{r^{\prime },u^{\prime }\}={\mathcal{R}}_b(\{r,u\})$$

By iterating this procedure, we can see how the parameters "flow" as we progressively zoom out to larger and larger length scales. This is called the Renormalization Group flow. The equations describing this flow are known as beta functions.

This framework allows us to understand critical phenomena in a profound way:

• Fixed Points: A critical point corresponds to a fixed point of the RG transformation, where the parameters stop changing: $\{r^{\ast },u^{\ast }\}={\mathcal{R}}_b(\{r^{\ast },u^{\ast }\})$. At a fixed point, the system is scale-invariant, which is the hallmark of criticality.
• Universality: Different physical systems, with different microscopic details (and thus different starting points in theory space), might flow towards the same fixed point. This explains universality: their critical behavior is identical because it is governed by the properties of that shared fixed point, not their microscopic origins.

The following reading provides a concrete application of this procedure to the simple Gaussian model (where $u=0$) and introduces the crucial classification of operators.

Conclusion

In this lesson, we have constructed the formal machinery for Wilson's momentum-space renormalization group. We now have a well-defined, three-step procedure to analyze the LGW field theory we derived previously.

Key Takeaways:

• The RG procedure operates on the LGW Hamiltonian in momentum space, where modes are separated by a UV cutoff $\Lambda$.
• The procedure consists of three steps:
  1. Integrate: Integrate out "fast" modes between $\Lambda /b$ and $\Lambda$ to get an effective Hamiltonian for the "slow" modes.
  2. Rescale Momenta: Rescale ${\mathbf{q}}^{\prime }=b\mathbf{q}$ to restore the momentum cutoff to $\Lambda$.
  3. Rescale Field: Rescale the field ${\varphi }^{\prime }={\zeta }^{-1}{\varphi }_{<}$ to restore the canonical form of the kinetic term.
• This three-step process defines an RG transformation ${\mathcal{R}}_b$, which generates a flow in the space of coupling constants ($r,u,\dots$).
• Fixed points of this flow correspond to scale-invariant theories and thus describe critical points.

Preview of the Next Lesson:

We have set up the game board and the rules. In the next lesson, we will make our first move. We will perform the momentum-shell integration and rescaling steps to derive the RG flow equations for ${\varphi }^4$ theory at one-loop order. This will involve carrying out the integral from Step 1 using a perturbative expansion in the coupling constant $u$, which will give us the famous beta functions that describe the flow towards and away from the Gaussian and Wilson-Fisher fixed points.