Real-Space RG for the 2D Ising Model

Hello! Welcome back to our journey into the Renormalization Group.

In our last lesson, we built a complete conceptual framework for understanding RG flows. We learned to classify fixed points as stable, unstable, or marginal based on the eigenvalues of the linearized flow, and we saw how these classifications give the fixed points their physical meaning: stable fixed points represent thermodynamic phases, while unstable fixed points correspond to critical points.

Today, we transition from theory to practice. Our goal is to apply this machinery to one of the most celebrated problems in statistical physics. As you know from your background in condensed matter physics, solving the 2D Ising model was a landmark achievement. While an exact solution via transfer matrix methods exists, it is quite involved. The RG provides an alternative, more physical, and surprisingly powerful approximate method to understand its critical properties.

Our learning outcome for this lesson is to perform a real-space RG calculation for the 2D Ising model on a square lattice. We will explore two famous approximation schemes to achieve this, deriving the recursion relations, finding the critical fixed point, and calculating a critical exponent.

The Challenge of Exact Real-Space RG in 2D

Let's start by understanding why an exact real-space RG transformation is so difficult in dimensions greater than one. In the 1D Ising model, when we summed over every other spin (decimation), the remaining spins were coupled in exactly the same way as the original ones, just with a new coupling constant $K^{\prime }$. The form of the Hamiltonian remained invariant.

This elegant closure breaks down in 2D. To see why, let's watch the beginning of the following lecture by Prof. Mehran Kardar.

As the video explains, when we sum over a spin on a square lattice, it's coupled to four neighbors. Integrating it out introduces effective interactions between all four of these neighbors. This generates not only a next-nearest-neighbor coupling but also a four-spin coupling term. If we were to repeat this process, even more complex, long-range, multi-spin interactions would proliferate. Our simple Hamiltonian, $H=-J{\sum }_{⟨i,j⟩}{\sigma }_i{\sigma }_j$, is not closed under this transformation.

To make progress, we must use an approximation that forces the renormalized Hamiltonian back into a manageable form. We will now explore two such influential approximation schemes.

Method 1: Decimation on a Square Lattice (Approximate)

Our first approach tackles the square lattice directly. We will perform the decimation as intended but then make a judicious approximation to keep the Hamiltonian simple. This method is detailed in the notes by Franz Schwabl.

The basic idea is to:

1. Start with a Hamiltonian that includes both nearest-neighbor (K) and next-nearest-neighbor (L) couplings. We anticipate that L will be generated, so we include it from the start.
2. Perform a decimation by summing over a sublattice of spins. This generates a new Hamiltonian with couplings $K^{\prime },L^{\prime },M^{\prime }$ (where $M^{\prime }$ is a four-spin coupling), and so on.
3. Approximate: We assume that the newly generated couplings like $M^{\prime }$ are small and can be neglected. We truncate the parameter space to just $(K,L)$, creating a closed set of recursion relations.

Let's summarize the key results from this analysis:

• Recursion Relations: The RG flow in the approximate parameter space $(K,L)$ is given by: $$\begin{gathered} K^{\prime }=2K^2+L \\ {L}^{\prime }=K^2 \end{gathered}$$
• Fixed Points: This system of equations has three fixed points:
  1. A stable "high-temperature" fixed point at $(K^{\ast },L^{\ast })=(0,0)$.
  2. A stable "low-temperature" fixed point at $(K^{\ast },L^{\ast })=(\infty ,\infty )$.
  3. A non-trivial unstable fixed point at $(K_c^{\ast },L_c^{\ast })=(1/3,1/9)$. As we learned, this corresponds to the critical point.
• Critical Exponent: By linearizing the flow around the critical fixed point, we find the relevant eigenvalue ${\lambda }_1=(2+\sqrt{10})/3\approx 1.72$. Since the length scale changes by $b=\sqrt{2}$ in this decimation, we can find the thermal exponent $y_t=1/\nu$: $${\lambda }_1=b^{y_t}=(\sqrt{2}{)}^{1/\nu }⟹\nu =\frac{\log \sqrt{2}}{\log {\lambda }_1}=\frac{\log 2}{2\log {\lambda }_1}\approx 0.638$$

This value is a significant improvement over the mean-field result $\nu =0.5$, though it differs from the exact 2D Ising result $\nu =1$. This showcases the power and the approximate nature of the method.

Method 2: The Migdal-Kadanoff Bond-Moving Approximation

Our second method is a clever and computationally simpler scheme developed by Migdal and Kadanoff. Instead of dealing with the proliferating bonds from decimation, the idea is to "move" them out of the way first, reducing the problem to a more manageable 1D calculation.

The procedure for a scaling factor of $b=2$ is as follows:

1. Bond Moving: Consider the vertical bonds in the middle of each 2x2 plaquette. Move these bonds to lie on top of the vertical bonds on the plaquette edges. The problem now looks like a set of 1D horizontal chains that are coupled by vertical bonds of double strength, $2K$.
2. Decimation: Sum over the spins on the double-strength vertical bonds. Since these are now just 1D chains, we can use the exact 1D decimation rule we know. This step eliminates the vertical connections, leaving us with a set of decoupled spins.
3. Repeat: The process is then repeated in the other direction (move horizontal bonds, decimate vertically) to complete the full transformation. A simpler, though more approximate, approach combines these into a single step.

The following problem set guides you through a variant of this procedure.

Let's walk through the logic of the solution you would have derived.

1. Effective 1D Problem: After moving the bonds, we have 1D chains with coupling strength $\tilde{K}=2K$.
2. Decimation: We decimate these chains with a scaling factor $b=2$. From our study of the 1D Ising model, the decimation recursion relation is $K^{\prime }=\frac{1}{2}\log (\cosh (2\tilde{K}))$.
3. Full Recursion Relation: Substituting $\tilde{K}=2K$, we get the Migdal-Kadanoff recursion relation for the 2D square lattice: $$K^{\prime }=\frac{1}{2}\log (\cosh (4K))$$
4. Fixed Points:
   • $K^{\ast }=0$ is a stable fixed point.
   • For large $K$, $K^{\prime }\approx 2K-\frac{1}{2}\log 2$, so the flow is towards $K\to \infty$, which is another stable fixed point.
   • The unstable critical fixed point $K_c^{\ast }$ is the non-zero solution to $K_c^{\ast }=\frac{1}{2}\log (\cosh (4K_c^{\ast }))$. A numerical solution gives $K_c^{\ast }\approx 0.304$. This is in the same ballpark as the exact $K_c\approx 0.44$.
5. Critical Exponent: We linearize the flow by calculating the derivative at the fixed point: $$\lambda =\frac{d{K}^{\prime }}{dK}{|}_{K_c^{\ast }}=2\tanh (4K_c^{\ast })$$ Plugging in the value for $K_c^{\ast }$, we get $\lambda \approx 1.678$. The length rescaling factor is $b=2$. Therefore: $$\lambda =b^{1/\nu }=2^{1/\nu }⟹\nu =\frac{\log 2}{\log \lambda }\approx \frac{0.693}{0.518}\approx 1.338$$

Again, the result is approximate but captures the essence of the critical behavior. It's fascinating that two different, plausible approximation schemes (Schwabl's decimation and Migdal-Kadanoff bond-moving) give different values for the exponents, underscoring that these are not exact methods.

Conclusion

In this lesson, we have rolled up our sleeves and performed real-space RG calculations for the 2D Ising model. We saw firsthand that exact decimation is intractable in higher dimensions due to the proliferation of new interaction terms. We then successfully implemented two distinct approximation schemes—a truncated decimation and the Migdal-Kadanoff bond-moving procedure—to derive recursion relations, locate the critical fixed point, and estimate the critical exponent $\nu$. Both methods yielded non-trivial results that improve upon mean-field theory, highlighting the power of the RG paradigm.

Key Takeaways:

• Direct decimation in $d>1$ is not a closed procedure; it generates an infinite hierarchy of new couplings.
• Real-space RG calculations therefore rely on approximations that truncate the parameter space of the Hamiltonian.
• The truncated decimation on the square lattice gave $\nu \approx 0.638$.
• The Migdal-Kadanoff bond-moving approximation provided a simpler calculation, yielding $\nu \approx 1.338$.
• The fact that different valid approximations give different exponents demonstrates that these methods are not exact but are powerful tools for gaining physical insight into critical phenomena.

Preview of the Next Lesson:

The approximations we used today, while insightful, are somewhat uncontrolled. We arbitrarily decided which bonds to move or which couplings to neglect. This raises a natural question: how can we be sure our results are meaningful? This limitation motivates a more systematic, but also more abstract, approach. In the next lesson, we will explore the concept of universality in the language of RG flows, understanding how different systems can flow to the same fixed point, and thus belong to the same universality class, regardless of their microscopic details. This will solidify the connection between RG flows and one of the deepest ideas in critical phenomena.