Universality, RG Flows, and Basins of Attraction

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Hello! Welcome to the next lesson in our exploration of the Renormalization Group.

In our last session, we performed real-space RG calculations for the 2D Ising model using different approximation schemes. We saw that while both methods gave us non-mean-field critical exponents, the specific values depended on the approximation used. This raises a profound question: If the results depend on the details of our calculation, how can we explain the phenomenon of universality, where the critical exponents are independent of the microscopic details of a system?

Today, we will answer this question. We will step back from specific calculations and look at the global structure of RG transformations. Our goal is to explain universality in the language of RG flows and basins of attraction. This conceptual framework is one of the crowning achievements of 20th-century physics, providing a deep and elegant explanation for one of nature's most striking patterns.

The Big Picture: Flows in Theory Space

First, we need to generalize our perspective. In our previous calculations, we tracked only one or two coupling constants, like $K$ and $L$ . However, a truly general Hamiltonian, consistent with the symmetries of a system like the Ising model (a $Z_{2}$ symmetry $ϕ \to - ϕ$ ), can contain an infinite number of terms: $ϕ^{2}, ϕ^{4}, ϕ^{6}, (\nabla ϕ)^{2}, ϕ^{2} (\nabla ϕ)^{2}$ , and so on. The space of all possible coupling constants for these terms forms an infinite-dimensional "theory space". The RG procedure—coarse-graining and rescaling—acts as a transformation that moves a point in this space, tracing out a trajectory or a "flow".

Let's begin with a high-level conceptual video that illustrates this idea of flows in parameter space and their convergence to fixed points.

Renormalization: Why Bigger is Simpler

This video beautifully explains the idea of coarse-graining, the resulting flow in a 'parameter space', and how this leads to simplification at large scales.

Watch from 10:50 to 14:01. Focus on how repeated coarse-graining and zooming out creates a 'renormalization group flow' in parameter space, and how these flows tend to converge towards fixed points or lines.

Now, let's formalize this picture. The following reading from David Tong's lecture notes lays out the mathematical framework for RG flows in theory space.

[PDF] 3. The Renormalisation Group

The set of all possible Hamiltonians compatible with a system's symmetries forms an infinite-dimensional space. The RG transformation induces a flow within this space. These notes provide a clear introduction to this powerful concept.

Please read Section 3.1, 'What's the Big Idea?'. Pay attention to how the free energy is generalized to include all possible terms (couplings). Follow the three steps of the RG procedure (integrate, rescale momenta, rescale fields) and understand how this process generates a flow in the space of coupling constants, as depicted in Figure 22.

To summarize, each point in theory space represents a specific Hamiltonian. The RG transformation, $R_{b}$ , maps one point (Hamiltonian) to another, describing the system at a larger length scale. Repeated application traces out a flow, showing how the effective theory changes as we zoom out.

Fixed Points and Their Stability

The destinations of these RG flows are the key. As the video suggested, flows often converge to special fixed points, which are points in theory space that are invariant under the RG transformation. A fixed point Hamiltonian $H^{*}$ satisfies $R_{b} (H^{*}) = H^{*}$ .

Fixed points describe scale-invariant systems, which must have a correlation length $ξ = 0$ or $ξ = \infty$ .

Trivial Fixed Points ( $ξ = 0$ ): These correspond to the completely ordered (zero temperature) or completely disordered (infinite temperature) phases.
Critical Fixed Points ( $ξ = \infty$ ): These describe systems at a critical point, exhibiting fluctuations on all length scales.

To understand universality, we must analyze the geometry of the flow near a critical fixed point. We linearize the RG transformation $S^{'} = R_{b} (S)$ around a fixed point $S^{*}$ :

S_{i}^{'} - S_{i}^{*} = j \sum T_{ij} (S_{j} - S_{j}^{*})

where $T_{ij} = \frac{\partial S _{i}^{'}}{\partial S _{j}}_{S^{*}}$ is the linearization matrix. The eigenvectors of this matrix, let's call them $O_{i}$ , are called scaling operators or eigenoperators. They represent fundamental directions of deformation away from the fixed point. The corresponding eigenvalues, $λ_{i} = b^{y_{i}}$ , tell us how the flow behaves along these directions.

This leads to a crucial classification:

Relevant Operators ( $y_{i} > 0 ⟹ λ_{i} > 1$ ): The flow is away from the fixed point along this direction. Adding a relevant operator to the fixed-point Hamiltonian is a significant perturbation that drastically changes the long-distance physics.
Irrelevant Operators ( $y_{i} < 0 ⟹ λ_{i} < 1$ ): The flow is towards the fixed point along this direction. The effect of such an operator diminishes as we zoom out, making it "irrelevant" to the critical behavior.
Marginal Operators ( $y_{i} = 0 ⟹ λ_{i} = 1$ ): The flow is tangential to the fixed point. Higher-order analysis is needed to determine if they are truly marginal or become weakly relevant/irrelevant.

Universality Explained: Basins of Attraction

We now have all the pieces to explain universality. The central idea is the basin of attraction of a fixed point: the set of all points in theory space that flow towards that fixed point.

The basin of attraction for a critical fixed point is also called the critical surface. This surface is the manifold spanned by all the irrelevant operators. Any Hamiltonian located on this surface will flow to the critical fixed point and thus describe a system that undergoes a continuous phase transition.

RG flow diagram showing the critical surface and relevant/irrelevant directions

This figure from Tong's notes illustrates the concept of a critical surface. The red line represents a relevant direction, along which the flow moves away from the fixed point (the black dot). The blue dotted lines represent irrelevant directions, along which the flow moves towards the fixed point, defining the critical surface (the basin of attraction). Any point on the blue surface flows to the critical point.

The following reading provides a superb, concise explanation of how these concepts come together to explain universality.

[PDF] Chapter 4 Renormalisation Group - TCM

With the concepts of fixed points and operator classification in hand, we can now give a precise meaning to universality. It is all about the geometry of the RG flows. These notes from Cambridge provide a formal and clear explanation.

Read the text associated with Figure 4.2. It explains how the basin of attraction defines the phase transition surface, how experimental trajectories intersect this surface, and finally, how this provides an explanation for universality. Pay close attention to the argument about co-dimension one, which explains why we typically only need to tune one parameter (like temperature) to reach a critical point.

The argument is as follows:

Consider different physical systems: the Ising model on a square lattice, the Ising model on a triangular lattice, a liquid-gas system near its critical point. These systems have vastly different microscopic Hamiltonians, meaning they start at different points in theory space.
However, they might share the same fundamental symmetries (e.g., $Z_{2}$ for Ising and liquid-gas) and dimensionality. This often means their starting points, though different, lie on the same critical surface.
Under the RG flow, the components of their Hamiltonians corresponding to irrelevant operators (microscopic details like lattice structure, exact coupling strengths) shrink and vanish.
All these different systems are driven by the flow along the irrelevant directions towards the same critical fixed point.
Critical exponents are determined by the eigenvalues ( $y_{i}$ ) of the relevant operators at the fixed point. Since all these systems flow to the same fixed point, they are governed by the same set of relevant operators and eigenvalues.
Therefore, they all exhibit the same critical exponents and belong to the same universality class.

This is the power of the RG explanation: universality arises because the long-distance physics is controlled by the properties of a shared destination (the fixed point), not the unique starting points (the microscopic Hamiltonians).

Let's watch one final clip that beautifully visualizes how the boundary between basins of attraction for different stable fixed points defines the phase transition itself.

Renormalization: Why Bigger is Simpler

This clip provides an excellent schematic of how the phase transition boundary emerges from the RG flow picture. It shows how slightly changing an external parameter like temperature can move your system from one basin of attraction to another, causing an abrupt change in macroscopic properties.

Watch from 14:01 to 15:43. This directly connects the RG flow diagram to the phase diagrams you are familiar with from thermodynamics.

Test your understanding!

Imagine a critical fixed point $S^{*}$ in a 3D Ising universality class. It has two relevant operators: one related to temperature ( $t$ ) with exponent $y_{t} > 0$ , and one related to the magnetic field ( $h$ ) with exponent $y_{h} > 0$ . All other infinite directions away from $S^{*}$ are irrelevant.

What is the "co-dimension" of the critical surface for this fixed point? What does this mean physically about how you would tune an experimental system to observe this critical point?
Suppose you have two completely different magnetic materials. Model A is a simple nearest-neighbor Ising model on a cubic lattice. Model B has nearest-neighbor, next-nearest-neighbor, and some complicated four-spin interactions. Both exhibit a ferromagnetic transition. According to the RG theory of universality, what must be true about the starting points of Model A and Model B in theory space for them to have the same critical exponents?
If you add a small interaction to Model A that breaks the up-down ( $σ \to - σ$ ) symmetry, what do you expect to happen to its RG flow? Will it still flow to the Ising fixed point $S^{*}$ ? Why or why not?

Show answer

The co-dimension of a surface is the dimension of the ambient space minus the dimension of the surface. Here, the number of relevant directions is 2. The critical surface is the subspace where the coefficients of these relevant operators are zero. So, the critical surface has co-dimension 2. Physically, this means you must tune two parameters to zero to land on the critical point. For a magnet, this corresponds to setting the external field $h = 0$ and tuning the temperature $T$ to the critical temperature $T_{c}$ (which makes the effective reduced temperature $t = T - T_{c} = 0$ ).
For Model A and Model B to have the same critical exponents, they must belong to the same universality class. In the language of RG, this means that their initial Hamiltonians, while being very different points in the infinite-dimensional theory space, must both lie on the same critical surface associated with the 3D Ising fixed point. Under the RG flow, the differences in their microscopic details (like the next-nearest-neighbor and four-spin couplings in Model B) correspond to irrelevant directions and will be "washed out," causing both systems to flow to the same critical fixed point $S^{*}$ .
The Ising fixed point $S^{*}$ is defined by, among other things, its $Z_{2}$ ( $σ \to - σ$ ) symmetry. An interaction that breaks this symmetry (e.g., a term proportional to $σ_{i}^{3}$ , which is just $σ_{i}$ ) introduces a perturbation that is not symmetric. This perturbation corresponds to a new direction in theory space—essentially the magnetic field direction. As we know the magnetic field is a relevant operator, adding this term will push the system off the $h = 0$ critical surface. The RG flow will no longer go to the critical fixed point $S^{*}$ . Instead, it will be driven away, likely towards a trivial low- or high-temperature fixed point, and the sharp phase transition will be destroyed (or "rounded"). This demonstrates that fundamental symmetries are a crucial property determining the universality class.

Conclusion

In this lesson, we have constructed a powerful conceptual framework for understanding universality. By viewing phase transitions through the lens of the Renormalization Group, the mystery of why disparate systems behave identically near criticality is elegantly resolved.

Key Takeaways:

The RG transformation generates a flow in an infinite-dimensional space of Hamiltonians ("theory space").
Critical points correspond to unstable fixed points of this flow, which are scale-invariant.
The flow near a fixed point is characterized by relevant, irrelevant, and marginal directions (operators).
The basin of attraction of a critical fixed point is its critical surface, the set of all Hamiltonians that flow to it.
Universality arises because many different microscopic Hamiltonians (different starting points) can lie on the same critical surface. The RG flow washes away the irrelevant microscopic details, causing all of them to flow to the same fixed point. Since critical exponents are determined by the properties (relevant eigenvalues) of the fixed point, all systems in its basin of attraction share the same exponents.

Preview of the Next Lesson:

We have now reached a deep conceptual understanding of why RG works. However, the real-space RG calculations we performed in the previous lesson were ad-hoc and approximate. We arbitrarily chose which bonds to move or which new couplings to ignore. This makes it difficult to systematically find the fixed points and calculate their properties. In our next lesson, we will discuss these limitations of real-space RG and see how they motivate a move to a more powerful and systematic framework: momentum-space RG applied to continuous field theories. This will pave the way for powerful techniques like the epsilon expansion.