RG and Crossover Phenomena

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Hello! Welcome to our lesson on crossover phenomena.

In our last session, we saw the impressive predictive power of the $ϵ$ -expansion by comparing its (resummed) predictions for the 3D Ising model with high-precision numerical results. Our focus was on the behavior at the Wilson-Fisher fixed point, which describes the universal properties of a system precisely at its critical point.

However, real physical systems are not always exactly at a critical point, and their microscopic details can be complex. This raises a natural question: how does a system's behavior evolve as it approaches criticality? And what happens if its parameters place it "between" different universality classes? This is the domain of crossover phenomena. Today, we will see how the renormalization group provides a beautiful and powerful framework for describing how a system can "cross over" from one type of behavior (e.g., mean-field) to another (e.g., Ising-like) as we change an external parameter like temperature or system size.

Our goal is to describe how the RG framework explains crossover phenomena between different critical behaviors. We will explore this through several key examples, from the fundamental crossover from mean-field theory to the more subtle effects of dimensionality and multicritical points.

1. The RG Flow Picture of Crossover

At its heart, the RG framework describes how the effective Hamiltonian of a system changes as we look at it on larger and larger length scales. This generates a "flow" in the space of all possible Hamiltonians. Fixed points of this flow correspond to scale-invariant theories and thus represent universality classes.

To get a visual intuition for this, let's watch a short video segment that illustrates the concept of RG flow and how it relates to phase transitions.

Renormalization: Why Bigger is Simpler

The following clip from the channel 'Broken Symmetries' provides an excellent schematic of RG flow, showing how different initial microscopic Hamiltonians (different starting points in parameter space) can flow towards different fixed points, leading to distinct macroscopic phases.

Please watch the segment from 14:14 to 16:26. Pay close attention to the diagram with two fixed points and the boundary (the separatrix) between their basins of attraction. This boundary is the key to understanding phase transitions and crossover.

As the video explains, a system's macroscopic properties are determined by which fixed point its RG trajectory flows towards. A crossover phenomenon occurs when a single RG trajectory is influenced by more than one fixed point. For a certain range of scales, the flow might pass near one fixed point, leading the system to exhibit that fixed point's characteristic behavior. But as the system is coarse-grained further (e.g., by getting closer to $T_{c}$ ), the trajectory may veer away and head towards a different, more stable fixed point, causing the system's behavior to "cross over".

2. Crossover from Mean-Field to Critical Behavior

The most fundamental example of crossover is the transition from mean-field behavior, which is valid far from the critical point, to the true critical behavior governed by the Wilson-Fisher fixed point. We have all the tools to understand this now.

Recall from our study of the $ϵ$ -expansion that for dimensions $d < 4$ , the RG flow for the $ϕ^{4}$ theory has two key fixed points:

The Gaussian Fixed Point (GFP): Located at $t = 0, u = 0$ . Here, the interaction term $u$ is irrelevant for $d > 4$ but becomes relevant for $d < 4$ . This fixed point describes mean-field theory.
The Wilson-Fisher Fixed Point (WFFP): Located at a non-zero interaction strength, $u^{*} > 0$ . This fixed point is stable for $d < 4$ and describes the true critical behavior of the Ising universality class.

The following reading details the RG flow equations and explains this crossover.

[PDF] Physics 7240: Advanced Statistical Mechanics Lecture 5

This section from Leo Radzihovsky's lecture notes analyzes the RG flow equations for φ⁴ theory. It explicitly demonstrates the crossover from the unstable Gaussian fixed point to the stable Wilson-Fisher fixed point.

Please read section '2. d < duc = 4 (ϵ > 0)' on pages 34-35. Focus on the description of the flow diagram (Figure 10) and the role of the Ginzburg length, ξG, which is introduced as the scale controlling the crossover.

As the notes describe, for any system with $d < 4$ , the Gaussian fixed point is unstable. Any initial Hamiltonian with a non-zero interaction $u$ will eventually flow away from the GFP towards the stable WFFP.

The crossover can be understood in terms of the Ginzburg length, $ξ_{G}$ , defined in the text as $ξ_{G} \sim (K^{2} / u)^{1/ (4 - d)}$ . This length scale divides two regimes of behavior:

Mean-Field Regime ( $ξ (T) ≪ ξ_{G}$ ): When the system's correlation length is much smaller than the Ginzburg length (i.e., when we are far from $T_{c}$ ), the RG flow has not had a chance to move far from the GFP. The system's behavior is well-described by mean-field theory (e.g., critical exponent $γ = 1$ ).
Critical Regime ( $ξ (T) ≫ ξ_{G}$ ): As the temperature approaches $T_{c}$ , the correlation length $ξ (T)$ grows. Once $ξ (T)$ exceeds $ξ_{G}$ , the RG flow is driven strongly towards the Wilson-Fisher fixed point. The system's behavior "crosses over" to that of the Ising universality class (e.g., $γ \approx 1.24$ in 3D).

In essence, the Ginzburg criterion, which you encountered earlier, finds its formal justification in the RG framework as a crossover scale between the Gaussian and Wilson-Fisher fixed points.

3. Dimensional Crossover

Another classic example of crossover occurs when a system's geometry restricts its effective dimensionality. Consider a 3D magnetic material shaped into a very thin film.

[PDF] Physics 7240: Advanced Statistical Mechanics Lecture 5

The same lecture notes also provide an excellent discussion of dimensional crossover using the concept of finite-size scaling.

Please read the subsection 'An important generalization...' within Section 3, 'Finite-size scaling', on pages 7-8. It specifically discusses the case of a 'film' of thickness Lz.

The physics is again governed by the competition between two length scales: the correlation length $ξ (T)$ and the film thickness $L_{z}$ .

3D Behavior ( $ξ (T) ≪ L_{z}$ ): Far from $T_{c}$ , correlations are much smaller than the film thickness. The system is unaware of its confinement and behaves like a 3D system. Its properties are governed by the 3D Wilson-Fisher fixed point.
2D Behavior ( $ξ (T) ≫ L_{z}$ ): As $T \to T_{c}$ , the correlation length grows. Once it becomes much larger than the film thickness, the correlations are effectively "squashed" into the two larger dimensions. The system's fluctuations and long-range order become characteristic of a 2D system.

The RG flow "crosses over" from being governed by the 3D WFFP to the 2D WFFP. The crossover temperature $T_{L}$ is the point where $ξ (T_{L}) \sim L_{z}$ . Since $ξ (T) \sim ∣ T - T_{c} ∣^{- ν_{3 D}}$ , this crossover occurs when $∣ T - T_{c} ∣ \sim L_{z}^{- 1/ ν_{3 D}}$ .

Test your understanding!

Consider a 3D Ising model confined to a long, thin wire with a square cross-section of side length $L$ .

Far from $T_{c}$ , what kind of critical behavior do you expect?
What happens as $T$ approaches $T_{c}$ ? At what temperature scale (in terms of $L$ and critical exponents) does the crossover occur?
What is the ultimate low-temperature critical behavior? (Hint: Recall our results for the 1D Ising model from Module 3).

Show answer

Far from $T_{c}$ : The correlation length $ξ (T) ≪ L$ . The system is not constrained by the wire's geometry and will exhibit standard 3D Ising critical behavior.
Approaching $T_{c}$ : As the temperature gets closer to $T_{c}$ , $ξ (T)$ grows. A crossover will occur when $ξ (T) \sim L$ . This happens at a temperature scale $∣ T - T_{c} ∣ \sim L^{- 1/ ν_{3 D}}$ . For temperatures closer to $T_{c}$ than this, where $ξ (T) ≫ L$ , the system will cross over to effectively 1D behavior.
Ultimate Behavior: The 1D Ising model does not have a phase transition at any finite temperature. Therefore, the finite cross-section $L$ will ultimately round off the phase transition. The susceptibility and specific heat, instead of diverging, will reach a finite peak whose height and position depend on $L$ . The system will not achieve true long-range order.

4. Advanced Example: Tricritical Crossover

Let's now consider a more complex scenario that showcases the full power of the RG framework. Some systems can exhibit a tricritical point, where a line of second-order (critical) transitions meets a line of first-order transitions. In the RG landscape, this means the Gaussian fixed point has (at least) two relevant directions, and a system's behavior depends sensitively on its initial parameters.

What happens if a system's microscopic Hamiltonian places it in the basin of attraction of the Wilson-Fisher (Ising) fixed point, but very close to the "tricritical surface" that separates it from the first-order region?

[PDF] Renormalization group domains of the scalar Hamiltonian

This research paper by Bagnuls and Bervillier uses a non-perturbative RG method to study the domains of attraction for different fixed points. It provides a beautiful illustration of a 'retarded' crossover from classical to Ising behavior.

Please read Section 3, 'Trajectories for u₄ < 0' (pages 565-569). Focus on the conceptual argument presented in Figure 4 and the explanation of Figure 5. The key idea is the 'retarded classical-to-Ising crossover' for trajectories that start near the tricritical surface (path B in Fig. 4).

The paper reveals a fascinating phenomenon. As illustrated in their Figure 4, an RG trajectory that starts near the tricritical surface (path B) will first be drawn towards the Gaussian fixed point. It spends a long "time" (i.e., over many steps of coarse-graining) flowing near the GFP before the relevant perturbation associated with the WFFP finally wins out, causing the trajectory to veer away and eventually flow to the stable Wilson-Fisher fixed point.

This has a direct experimental consequence, shown in Figure 5 of the paper:

As you approach $T_{c}$ , you will first measure effective critical exponents that are characteristic of the Gaussian fixed point (i.e., mean-field values like $ν = 0.5$ ).
Only when you get extremely close to the true critical temperature will the system's behavior cross over to the true Ising critical exponents (e.g., $ν \approx 0.63$ ).

This "retarded crossover" is a purely dynamical effect of the RG flow. It demonstrates how the global structure of the flow in parameter space, not just the properties of a single fixed point, dictates the rich phenomenology of critical systems.

Conclusion

Today we have explored how the Renormalization Group provides a unified and powerful language to describe crossover phenomena. Instead of seeing different behaviors (mean-field, 3D Ising, 2D Ising) as separate theories, RG shows them to be different regimes of a single, unified flow in parameter space.

Key Takeaways:

Crossover occurs when a system's RG trajectory is influenced by more than one fixed point. The system exhibits different universal behaviors on different length scales (or in different temperature ranges).
The crossover from mean-field to critical behavior is understood as the RG flow being repelled by the unstable Gaussian fixed point and attracted to the stable Wilson-Fisher fixed point, with the Ginzburg length setting the crossover scale.
Dimensional crossover happens in finite systems when the correlation length becomes comparable to the system size, causing the effective dimensionality to change.
Complex crossovers, such as near a tricritical point, can lead to a sequence of different behaviors, like a "retarded" crossover from classical to Ising exponents, as the RG flow navigates a more complex landscape.

Preview of the Next Lesson:

Throughout this course, we have almost exclusively focused on the Ising model, which possesses a discrete, up/down ( $Z_{2}$ ) symmetry. What happens if the order parameter has a continuous symmetry, like the 2D vectors in the XY model or the 3D vectors in the Heisenberg model? In our next lesson, we will begin our exploration of advanced topics by studying the Mermin-Wagner theorem, a profound result that places strong constraints on the existence of spontaneous symmetry breaking in low-dimensional systems with continuous symmetries.