Scale Invariance and Conformal Symmetry in Critical Systems

From Phase Transitions to Scale Invariance

Many-body systems can exist in different macroscopic phases. A familiar example is water, which can be a liquid or a gas. The transition between these phases can be abrupt (first-order), like boiling water at standard pressure, or continuous (second-order). We are interested in the latter.

At a critical point, a system undergoes a continuous phase transition. At this precise point, the system exhibits remarkable properties. Perhaps the most important is the divergence of the correlation length, $\xi$.

The correlation length measures the typical distance over which fluctuations in the system are correlated. For example, in a magnet, it's the distance over which the orientation of one spin significantly influences another.

• Away from the critical point, correlations decay exponentially with distance, e.g., $⟨s(x)s(y)⟩\sim e^{-|x-y|/\xi }$, and $\xi$ is finite.
• At the critical point, the correlation length diverges: $\xi \to \infty$. The system loses its characteristic length scale.

What does it mean for a system to have no preferred length scale? It means the system looks the same at all scales of observation. Fluctuations occur on all length scales simultaneously. This property is called scale invariance. At a critical point, correlation functions no longer decay exponentially but instead follow power laws, a hallmark of scale invariance.

From Scale Invariance to Conformal Invariance

The emergence of scale invariance is a profound simplification. However, in many physical systems, the symmetry at the critical point is even larger. It is often enhanced from simple scale invariance to full conformal invariance.

Recall that scale transformations, $x^{\prime \mu }=\lambda x^{\mu }$, are a subset of conformal transformations. Conformal transformations are angle-preserving coordinate changes. While a scale transformation resizes everything uniformly, a general conformal transformation can locally stretch and rotate the space in a position-dependent way, as long as angles between intersecting lines remain unchanged.

Why should scale invariance imply this larger symmetry? The argument is subtle but powerful and relies on the properties of the stress-energy tensor $T_{\mu \nu }$. Your background in QFT will be helpful here.

1. Scale Invariance and the Dilatation Current: In a scale-invariant field theory, there is a conserved current associated with scale transformations (dilatations), called the dilatation current, $J_D^{\mu }=x_{\nu }{T}^{\mu \nu }$.
2. Conservation and the Stress Tensor Trace: The conservation of this current, ${\partial }_{\mu }{J}_D^{\mu }=0$, implies (after a short calculation using ${\partial }_{\mu }{T}^{\mu \nu }=0$) that the trace of the stress-energy tensor must vanish: $T_{\mu }^{\mu }=0$.
3. Traceless Stress Tensor and Conformal Symmetries: For a generic symmetry generated by a vector field ${ϵ}^{\mu }(x)$, the corresponding current $J_{ϵ}^{\mu }={ϵ}_{\nu }{T}^{\mu \nu }$ is conserved if ${\partial }_{\mu }{ϵ}_{\nu }+{\partial }_{\nu }{ϵ}_{\mu }=0$ (the Killing equation). However, if $T_{\mu }^{\mu }=0$, the condition for current conservation relaxes to ${\partial }_{\mu }{ϵ}_{\nu }+{\partial }_{\nu }{ϵ}_{\mu }=\frac{2}{d}{g}_{\mu \nu }(\partial \cdot ϵ)$ (the conformal Killing equation).

This relaxed condition allows for more solutions than just the Poincaré symmetries and dilatations. It allows for the special conformal transformations (SCTs) we studied in the first module. Thus, in a theory where the stress-energy tensor is traceless, scale invariance naturally leads to full conformal invariance.

While the implication "scale invariance $⟹$ conformal invariance" is not universally guaranteed, it has been proven for unitary, Lorentz-invariant QFTs in dimensions $d=2$ and $d=4$. In other dimensions, it remains a well-supported conjecture that holds in all known examples.

Critical Universality: The Domain of CFT

We've established that the long-distance physics of a system at its critical point is described by a scale-invariant—and typically conformal—field theory. The final piece of the puzzle is the remarkable phenomenon of universality.

It turns out that many microscopically different systems exhibit the exact same behavior at their critical points. For instance:

• Water at its liquid-gas critical point.
• The 3D Ising model (a lattice of classical spins) at its critical temperature.
• A uniaxial magnet at its Curie temperature.

All of these systems, despite their disparate microscopic physics, are described by the same set of critical exponents and, more deeply, by the same Conformal Field Theory at long distances. They are said to belong to the same universality class.

This is the power of our approach. By studying a single CFT (e.g., the "3D Ising CFT"), we are simultaneously studying the universal critical behavior of a vast class of physical systems. The abstract constraints we derive from the conformal bootstrap have direct, measurable consequences in laboratories studying any of these systems.

Conclusion

Today, we have built the crucial bridge from the abstract world of CFT to the physical realm of statistical mechanics. We have seen how the seemingly complex and diverse world of critical phenomena simplifies into the elegant and constrained structure of Conformal Field Theory.

Key Takeaways:

• Systems at a continuous (second-order) phase transition are located at a critical point.
• At a critical point, the correlation length $\xi$ diverges, meaning the system loses its intrinsic scale and becomes scale-invariant.
• In many local, unitary quantum field theories, scale invariance implies that the stress-energy tensor is traceless ($T_{\mu }^{\mu }=0$), which in turn enhances the symmetry to full conformal invariance.
• The phenomenon of universality means that diverse microscopic systems can belong to the same universality class, whose long-distance critical behavior is described by a single, unique CFT.

In our next lesson, we will make this connection even more concrete. We will see how the abstract scaling dimensions $\Delta$ of primary operators in a CFT are precisely the critical exponents that characterize the power-law behavior of physical observables near a critical point.