RG Flow, Fixed Points, and Critical Behavior

Hello! Welcome back to our study of the Renormalization Group.

In our last lesson, we performed a concrete RG calculation for the 1D Ising model using decimation. We derived the flow equation for the coupling constant, $K^{\prime }=\frac{1}{2}\ln (\cosh (2K))$, and used it to prove that a phase transition is impossible at any finite temperature. We saw that for any starting temperature $T>0$, the system always flows towards the high-temperature, disordered fixed point at $K^{\ast }=0$.

Today, we will generalize this picture. Our goal is to formalize the ideas we encountered and build a complete conceptual framework. We will move from the one-dimensional flow of a single parameter, $K$, to the concept of RG flow in a multi-dimensional parameter space. We will formally define what fixed points are, classify them based on their stability, and establish their profound connection to thermodynamic phases and the nature of critical points.

From a Single Coupling to a Parameter Space

The RG procedure, whether it's decimation or a more general Kadanoff block-spin transformation, acts on a system's Hamiltonian. The Hamiltonian is defined by a set of parameters: couplings, external fields, chemical potentials, etc. Let's denote this set of parameters as a vector $S=\{K_1,K_2,\dots \}$ that defines a point in a (potentially infinite-dimensional) parameter space.

The RG transformation, $R_b$, is a map that takes one Hamiltonian to another, and therefore one point in parameter space to another:
$S^{\prime }=R_b(S)$
Repeating this process traces out a trajectory, or an RG flow, in this space. This flow tells us how the effective interactions in our system change as we observe it on progressively larger length scales.

The following video provides an excellent conceptual bridge from the 1D model we studied to a more general flow in a 2D parameter space.

Fixed Points and Their Physical Meaning

As the video illustrates, the trajectories in this parameter space are not random; they have a distinct structure. Certain points are special: the flow either leads towards them or away from them. These are the fixed points of the RG transformation.

A fixed point $S^{\ast }$ is a point in parameter space that is invariant under the RG transformation:
$R_b(S^{\ast })=S^{\ast }$
Physically, this means the system described by the Hamiltonian $H(S^{\ast })$ is statistically self-similar or scale-invariant. It looks the same at all length scales.

What does this imply for the correlation length $\xi$? The RG transformation involves a length rescaling $x\to x/b$, which means the correlation length of the new system is $\xi (S^{\prime })=\xi (S)/b$. At a fixed point, this becomes:
$\xi (S^{\ast })=\frac{\xi (S^{\ast })}{b}$
Since the rescaling factor $b>1$, this equation has only two possible solutions for $\xi (S^{\ast })$:

1. $\xi (S^{\ast })=0$: This describes a system with no correlations beyond the microscopic lattice scale. These are trivial fixed points that represent stable thermodynamic phases, such as a completely disordered paramagnet at high temperature ($T=\infty$) or a perfectly ordered ferromagnet at zero temperature ($T=0$).

2. $\xi (S^{\ast })=\infty$: This describes a system with correlations at all length scales—the defining characteristic of a critical point. These are the crucial non-trivial fixed points.

The next segment of the video gives a superb qualitative explanation of how to identify these different types of fixed points from the direction of the RG flow.

Formalizing the Framework

We now have a powerful qualitative picture. Let's make it more rigorous. The flow of parameters towards or away from a fixed point defines its stability.

• Stable Fixed Points: Flows in the neighborhood of the fixed point are attracted towards it. These fixed points have a basin of attraction. All Hamiltonians whose parameters start within this basin will flow to the same stable fixed point, and thus describe the same macroscopic phase.
• Unstable Fixed Points: Flows move away from the fixed point in at least one direction. These correspond to critical points. To stay at an unstable fixed point, the system parameters must be tuned perfectly. Any infinitesimal deviation will cause the RG flow to carry the system away into one of the stable phases.

The set of all points in parameter space that flow into an unstable critical fixed point is called the critical surface. This surface is the phase boundary in the parameter space.

The following text formalizes these ideas beautifully.

This formalism provides a geometric explanation for universality. All systems whose parameters lie on the same critical surface flow to the same critical fixed point. Since the critical behavior is determined by the properties of the flow near this fixed point, all these systems share the same critical exponents, regardless of their microscopic details. They belong to the same universality class.

An Example: The 2D Ising Model

To solidify these concepts, let's look at the 2D Ising model. Unlike the 1D case, it has a finite-temperature phase transition, so its RG flow diagram must contain a non-trivial critical fixed point. An exact real-space RG is intractable because the decimation procedure generates an infinite number of new couplings (next-nearest-neighbor, four-spin interactions, etc.). However, an approximate calculation, where we truncate the parameter space, can still reveal the essential physics.

The flow diagram (Fig. 7.8 in the text) for the approximate 2D Ising model shows exactly the structure we've been discussing:

• A stable high-temperature fixed point at $(K,L)=(0,0)$. This governs the paramagnetic phase.
• A stable low-temperature fixed point at $(K,L)=(\infty ,\infty )$. This governs the ferromagnetic phase.
• An unstable critical fixed point at a finite $(K_c^{\ast },L_c^{\ast })$.

The line that separates the basins of attraction of the two stable fixed points is the critical surface. Any trajectory starting on this surface flows towards the critical fixed point. A physical system, like the standard Ising model with $L=0$, is represented by a line in this space parameterized by temperature ($K=J/k_{B}T$). The phase transition occurs at the temperature $T_c$ where this line intersects the critical surface.

Conclusion

In this lesson, we have constructed the core conceptual framework of the renormalization group. By moving from a concrete calculation to a general, geometric picture, we can now see how the RG provides a unified explanation for phase transitions, critical phenomena, and universality.

Key Takeaways:

• The RG transformation induces a flow in the space of Hamiltonian parameters.
• Fixed points of this flow correspond to scale-invariant systems. Trivial fixed points ($\xi =0$) describe stable thermodynamic phases (ordered or disordered), while non-trivial, unstable fixed points ($\xi =\infty$) describe critical points.
• The critical surface is the set of all points that flow into a critical fixed point, acting as the boundary between phases.
• Universality arises because all systems whose parameters lie on the same critical surface flow to the same critical fixed point and thus share the same critical behavior.

Preview of the Next Lesson:

We have qualitatively discussed the stability of fixed points based on the direction of flows. In our next lesson, we will make this analysis quantitative. We will learn how to linearize the RG transformation near a fixed point and show how the eigenvalues of this linearized map are directly related to the critical exponents of the phase transition.