Fixed Points and Stability in RG Flow

In our previous lesson, we developed the machinery to generate Renormalization Group flows, deriving recursion relations for both a discrete spin system and a continuous field theory. We saw that the act of coarse-graining—integrating out short-distance degrees of freedom—systematically alters the parameters of our theory. Now, we will analyze the structure of these flows to extract their profound physical meaning.

This lesson focuses on the central objects in any RG analysis: the fixed points. These are special points in the "theory space" of all possible Hamiltonians that remain unchanged by the RG transformation. By identifying these points and studying how the RG flow behaves in their vicinity, we can understand the long-distance fate of a system and predict its phase structure. You will learn how to find fixed points and, crucially, how to analyze their stability, which is the key to unlocking the secrets of critical phenomena.

What are Fixed Points?

A fixed point of an RG transformation represents a scale-invariant theory. Since the RG procedure involves "zooming out," a system at a fixed point looks statistically the same at all length scales. This can mean one of two things:

Trivial Fixed Points: The system is completely uncorrelated, with a correlation length $ξ = 0$ . This corresponds to a completely ordered state (like at zero temperature, $T = 0$ ) or a completely disordered state (like at infinite temperature, $T = \infty$ ).
Critical Fixed Points: The system has correlations on all length scales, meaning the correlation length is infinite, $ξ = \infty$ . This is the situation at a continuous phase transition.

Formally, if our RG transformation is a discrete map on the couplings $K \to K^{'} = f (K)$ , a fixed point $K^{*}$ satisfies $K^{*} = f (K^{*})$ . If the flow is continuous, described by a set of differential equations called beta functions, $\frac{d g _{i}}{d s} = β_{i} ({g_{j}})$ , a fixed point ${g_{j}^{*}}$ is a point where all beta functions vanish: $β_{i} ({g_{j}^{*}}) = 0$ for all $i$ .

To get a feel for how these flows are structured, the following video provides a qualitative picture for a generic system like the Ising model.

Renormalisation group

This video from Jos Thijssen's course on Statistical Mechanics illustrates the concept of an RG flow diagram in the space of coupling constants. It provides the crucial intuition for why flows behave the way they do.

Watch the segment from this qualitative analysis. The lecturer sketches a flow diagram in a two-dimensional parameter space (e.g., nearest-neighbor and next-nearest-neighbor couplings). Pay attention to the physical arguments for why the flow moves towards higher temperatures in the high-T phase and towards lower temperatures in the low-T phase. This leads to the identification of a high-temperature fixed point (disordered), a low-temperature fixed point (ordered), and a critical fixed point separating them.

Stability Analysis: The 1D Ising Model Revisited

Having established the concept, let's analyze the stability of the fixed points we can find. In the previous lesson, you derived the recursion relation for the 1D Ising model under decimation:

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

where $K = J / k_{B} T$ . The fixed points are the solutions to $K^{*} = \frac{1}{2} ln (cosh (2 K^{*}))$ . By inspection or graphical analysis, we find two solutions:

High-Temperature Fixed Point: $K^{*} = 0$ (corresponding to $T \to \infty$ )
Low-Temperature Fixed Point: $K^{*} \to \infty$ (corresponding to $T = 0$ )

But are these fixed points attractive or repulsive? That is, if we start near a fixed point, does the flow take us closer or push us away? To answer this, we linearize the flow equation around a fixed point $K^{*}$ . Let $K_{n} = K^{*} + δ K_{n}$ . After one RG step:

K_{n + 1} = f (K_{n}) = f (K^{*} + δ K_{n}) \approx f (K^{*}) + f^{'} (K^{*}) δ K_{n} = K^{*} + f^{'} (K^{*}) δ K_{n}

So, the deviation from the fixed point transforms as $δ K_{n + 1} \approx f^{'} (K^{*}) δ K_{n}$ . The stability is governed by the derivative $f^{'} (K^{*})$ :

If $∣ f^{'} (K^{*}) ∣ < 1$ , the deviation shrinks with each step. The fixed point is stable (an attractor).
If $∣ f^{'} (K^{*}) ∣ > 1$ , the deviation grows. The fixed point is unstable (a repeller).

The following video performs this exact analysis for the 1D Ising model.

13. Position Space Renormalization Group, Part 1

This lecture from MIT OpenCourseWare, which you've seen before, now becomes central. Having set up the RG for the 1D Ising model, the lecturer now analyzes its fixed points.

Watch the segment from this analysis. The lecturer finds the recursion relation for the coupling k (equivalent to our K) and analyzes its behavior near the fixed points k=0 and k=\infty. Notice how he linearizes the flow to show that k=0 is a stable fixed point, while k=\infty is unstable. He also introduces a change of variables to t = \tanh(k), which simplifies the recursion to t' = t^2, making the flow structure transparent.

The result of this analysis is profound. Since the only stable fixed point is the one at $K = 0$ (infinite temperature), any 1D Ising system with a finite initial coupling $K > 0$ will inevitably flow towards the disordered state under coarse-graining. This means the system looks disordered at large scales, regardless of the initial temperature (as long as it's not absolute zero). This is the RG explanation for why the 1D Ising model does not have a finite-temperature phase transition.

Stability Analysis in Field Theory

The same logic applies to field theories, but the formalism is slightly different. Instead of a discrete map, we have a continuous flow governed by beta functions. To analyze the stability of a fixed point ${g_{j}^{*}}$ , we linearize the beta functions:

\frac{d}{d s} (g_{i}^{*} + δ g_{i}) = β_{i} ({g_{j}^{*} + δ g_{j}}) \approx = 0 β_{i} ({g_{j}^{*}}) + j \sum \frac{\partial β _{i}}{\partial g _{j}}_{{g_{j}^{*}}} δ g_{j}

This gives a system of linear differential equations for the deviations $δ g_{i}$ :

\frac{d}{d s} (δ g_{i}) = j \sum M_{ij} δ g_{j} where M_{ij} = \frac{\partial β _{i}}{\partial g _{j}}_{{g_{j}^{*}}}

The solutions are of the form $δ g_{i} (s) = \sum_{k} c_{k} v_{i}^{(k)} e^{λ_{k} s}$ , where $λ_{k}$ and $v^{(k)}$ are the eigenvalues and eigenvectors of the matrix $M$ . The stability is determined by the sign of the eigenvalues $λ_{k}$ :

$λ_{k} > 0$ : The perturbation grows. The corresponding eigenvector direction is relevant.
$λ_{k} < 0$ : The perturbation shrinks. The corresponding eigenvector direction is irrelevant.
$λ_{k} = 0$ : The perturbation does not change at linear order. The direction is marginal.

The following reading formalizes this procedure and introduces the crucial concept of scaling dimensions.

Chapter 4 Renormalisation Group

This section from a textbook by B.D. Simons provides a concise, formal overview of the stability analysis around a fixed point.

Read section 4.2, titled Formal Approach. Focus on steps 4 and 5. The text explains how to linearize the recursion relations and introduces the eigenvalues \lambda_i(b) = b^{y_i}, where the exponents y_i are the scaling dimensions that determine stability. An operator is relevant if y_i > 0, irrelevant if y_i < 0, and marginal if y_i = 0.

Case Study 1: The Gaussian Fixed Point

The simplest fixed point in a field theory is the Gaussian fixed point, where all interaction couplings are zero. Let's analyze its stability for our $ϕ^{4}$ theory. We consider a general Hamiltonian with operators $ϕ^{n}$ . At the Gaussian fixed point, the fields rescale in a simple way, which allows us to determine the flow of the couplings $g_{n}$ for these operators.

3. The Renormalisation Group

This section from David Tong's lecture notes on statistical field theory provides a clear analysis of the Gaussian fixed point.

Please read the subsection In the Vicinity of the Fixed Point. The author analyzes the scaling of generic interaction terms \int d^d x \, g_n \phi^n around the free theory. The key result is the flow equation for the coupling: g_n(\zeta) = \zeta^{(1-n/2)d+n} g_{0,n} where \zeta is the rescaling factor. Pay close attention to the analysis of the \phi^4 coupling, g_4. The author shows its flow depends on the spatial dimension d, revealing that d=4 is a special "upper critical dimension".

The analysis shows that the scaling exponent for the $ϕ^{4}$ coupling is $4 - d$ .

For $d > 4$ , the exponent is negative. The $ϕ^{4}$ interaction is irrelevant. The flow drives it to zero, meaning the system becomes free at long distances. Mean-field theory, which neglects fluctuations, works well here.
For $d < 4$ , the exponent is positive. The $ϕ^{4}$ interaction is relevant. The flow drives it away from zero. The Gaussian fixed point is unstable. The system flows to a new, interacting fixed point.
This instability of the Gaussian fixed point below $d = 4$ is the reason mean-field theory fails to predict the correct critical exponents for systems like the 3D Ising model.

Case Study 2: The Wilson-Fisher Fixed Point

If the flow for $d < 4$ runs away from the Gaussian fixed point, where does it go? Wilson and Fisher showed that it flows to a new, non-trivial fixed point, now named in their honor. This fixed point exists in $d = 4 - ϵ$ dimensions, where $ϵ$ is small. The fact that the fixed-point coupling $g^{*}$ is proportional to $ϵ$ makes it accessible via perturbation theory.

3. The Renormalisation Group

This section, also from Tong's notes, analyzes the stability of this new, interacting fixed point.

Read the subsection The Wilson-Fisher Fixed Point. The text presents the beta functions for the mass (\mu^2) and coupling (\tilde{g}) in d=4-\epsilon dimensions. It finds the coordinates of the new fixed point (\mu_*^2, \tilde{g}_*) and then linearizes the flow around it.

The crucial result of this analysis is the stability matrix and its eigenvalues:

\frac{d}{d s} (δ μ^{2} δ g ~) = (2 - ϵ /3 0 \dots - ϵ) (δ μ^{2} δ g ~)

The eigenvalues are the diagonal entries: $λ_{t} = 2 - ϵ /3 > 0$ and $λ_{g} = - ϵ < 0$ . This means the Wilson-Fisher fixed point has one relevant direction (associated with the mass term, or temperature) and one irrelevant direction (associated with the $ϕ^{4}$ coupling).

This is a beautiful and powerful result. It tells us that to reach this critical point, we only need to tune one parameter in our experiment (e.g., the temperature). The system will then flow automatically to the fixed point value for all other irrelevant parameters (like the bare coupling strength), explaining the phenomenon of universality, which we will discuss in depth in a future lesson.

Conclusion

This lesson has been a deep dive into the heart of the RG method. We have moved from generating flows to interpreting them.

Key Takeaways:

Fixed points represent scale-invariant theories and correspond to either trivial (uncorrelated) states or critical points.
The stability of a fixed point determines the long-distance behavior of the system.
Stability is analyzed by linearizing the RG flow around the fixed point and finding the eigenvalues of the transformation. Positive eigenvalues correspond to unstable (relevant) directions, while negative eigenvalues correspond to stable (irrelevant) directions.
For the 1D Ising model, the only stable fixed point is at infinite temperature, explaining the absence of a phase transition.
For $ϕ^{4}$ theory, the Gaussian fixed point is unstable for dimensions $d < 4$ .
The flow for $d < 4$ is controlled by the interacting Wilson-Fisher fixed point, which has one relevant and one irrelevant direction, providing the foundation for understanding critical phenomena and universality.

In our next lesson, we will formalize the classification of operators as relevant, irrelevant, or marginal, and explore the profound physical implications of this classification.

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