Critical Exponents and CFT Scaling Dimensions

Hello! Welcome back to our exploration of Conformal Field Theory.

In our last lesson, we established a profound connection: the physics of a statistical system at a continuous phase transition, where its correlation length diverges, is described by a Conformal Field Theory. This happens because the absence of a characteristic length scale ( $ξ \to \infty$ ) leads to scale invariance, which in local, unitary theories is typically enhanced to full conformal invariance.

Today, we will make this connection even more quantitative and powerful. Our learning objective is to identify critical exponents with the scaling dimensions of operators in the corresponding CFT. You'll see how the abstract scaling dimensions ( $Δ$ ), which we introduced in the first module as fundamental data of a CFT, are directly related to experimentally measurable quantities that characterize a phase transition.

The Two Sides of the Dictionary: Exponents and Dimensions

To build our "dictionary," we first need to be precise about the terms on both sides: critical exponents on the statistical mechanics side, and scaling dimensions on the CFT side.

Critical Exponents

As we discussed, near a critical point, various thermodynamic quantities exhibit singular, power-law behavior. This behavior is characterized by a set of universal numbers called critical exponents.

Conformal Field Theory and Gravity

Let's begin with a formal list of the main critical exponents. This excerpt from the textbook Conformal Field Theory and Gravity by João Penedones provides a concise summary.

Please read section 1.1.1, "Critical exponents" on page 3. This short section defines the six key exponents ((\alpha, eta, \gamma, \delta, u, \eta)) that quantify the singular behavior of quantities like specific heat, magnetization, and correlation functions.

These exponents ( $α, β, γ, δ, ν, η$ ) are the "words" on the statistical mechanics side of our dictionary. They are, in principle, measurable in a lab experiment or a numerical simulation of a system like the Ising model.

Scaling Dimensions

On the CFT side, the fundamental objects are local operators $O_{i} (x)$ . As we saw in Module 1, primary operators transform simply under a dilatation $x \to λ x$ :

O_{i} (x) \to λ^{Δ_{i}} O_{i} (λ x)

The number $Δ_{i}$ is the scaling dimension of the operator $O_{i}$ . This dimension governs the behavior of correlation functions. At a critical point, which corresponds to a CFT, the two-point function of a scalar primary operator takes a universal power-law form dictated entirely by its dimension:

⟨ O_{i} (x) O_{i} (y)⟩ = \frac{C _{ii}}{∣ x - y ∣ ^{2 Δ_{i}}}

where $C_{ii}$ is a normalization constant.

This concept connects the abstract CFT operators to the underlying lattice model. A scaling operator in the CFT isn't just one microscopic variable on the lattice; it's a specific combination of them that behaves cleanly under coarse-graining.

Conformal Field Theory and Statistical Mechanics

John Cardy's lecture notes provide an excellent perspective on how these continuum scaling operators emerge from a lattice model. This will reinforce the physical meaning of scaling dimensions.

Please read the first page of Section 2, "Scale invariance" (page 4). Focus on equation (1) and the surrounding text, which defines scaling dimensions, denoted x_j, as the exponents required for multiplicative renormalization of lattice observables to achieve a well-defined continuum limit.

The scaling dimension is the "magic number" that reveals the scale-invariant structure. When physical quantities are rescaled by the system size raised to the power of the scaling dimension, data from different system sizes collapse onto a single universal curve.

Data Collapse in a Critical System — This plot is a beautiful experimental demonstration of scaling. A rescaled correlation function is plotted against a rescaled distance for various system sizes $L$. The fact that all data points fall on a single curve is called \"data collapse.\" This only happens when the correct scaling exponent, $x_\sigma$, which is the scaling dimension $\Delta_\sigma$, is used.

Translating the Dictionary: A Direct Example

Let's make the first and most direct translation. The critical exponent $η$ is, by definition, related to the decay of the two-point function of the order parameter (e.g., the local spin/magnetization $σ (x)$ ) precisely at the critical point:

⟨ σ (x) σ (0)⟩ \sim \frac{1}{∣ x ∣ ^{d - 2 + η}}

In the language of CFT, the order parameter $σ (x)$ is a scalar primary operator with some scaling dimension $Δ_{σ}$ . Its two-point function must therefore behave as:

⟨ σ (x) σ (0)⟩ \sim \frac{1}{∣ x ∣ ^{2 Δ_{σ}}}

By simply comparing the exponents of these two expressions, we arrive at our first dictionary entry:

2 Δ_{σ} = d - 2 + η ⟺ η = 2 Δ_{σ} - d + 2

This is a remarkable result. An experimentally measurable critical exponent, $η$ , is determined by a fundamental parameter, $Δ_{σ}$ , of the underlying abstract CFT.

Conformal Field Theory and Gravity

The following text from Penedones' notes formalizes this connection between the power-law decay of correlation functions and scaling dimensions.

Please read section 1.2.4, "Scaling operators" (pages 7-8). Pay close attention to the derivation leading to equation (1.49), which shows how scale covariance at a fixed point forces the two-point function to have a power-law decay determined by the scaling dimension \Delta_i.

The Full Dictionary: Linking All Exponents to Two Dimensions

The connection goes much deeper. In fact, for a system like the Ising model with a $Z_{2}$ symmetry ( $σ \to - σ$ ), all six critical exponents are determined by the scaling dimensions of just two primary operators:

The spin operator $σ (x)$ : The order parameter, which is odd under the $Z_{2}$ symmetry. Its dimension is $Δ_{σ}$ .
The energy operator $ϵ (x)$ : The operator that corresponds to the energy density. It is the leading $Z_{2}$ -even scalar operator. Its dimension is $Δ_{ϵ}$ .

The derivations for the other exponents involve the Renormalization Group (RG), which we will study in detail in the next lesson. The core idea is that moving away from the critical temperature, $t = (T - T_{c}) / T_{c} \neq = 0$ , corresponds to perturbing the CFT action by the energy operator: $S = S_{C F T} + t \int d^{d} x ϵ (x)$ . The response of the system to this perturbation depends on the dimension $Δ_{ϵ}$ . Similarly, applying a magnetic field $h$ corresponds to perturbing by the spin operator: $S = S_{C F T} + h \int d^{d} x σ (x)$ .

Without going through the full RG derivation now, we can state the results. This is our complete CFT-to-Criticality dictionary:

Critical Exponent	Expression in terms of Scaling Dimensions
$α$ (Specific Heat: $C \sim	t
$β$ (Magnetization: $M \sim (- t)^{β}$ )	$β = \frac{Δ _{σ}}{d - Δ _{ϵ}}$
$γ$ (Susceptibility: $\chi \sim	t
$δ$ (Critical Isotherm: $M \sim h^{1/ δ}$ )	$δ = \frac{d - Δ _{σ}}{Δ _{σ}}$
$ν$ (Correlation Length: $\xi \sim	t
$η$ (Correlation Function: $⟨ σ σ ⟩ \sim r^{- (d - 2 + η)}$ )	$η = 2 Δ_{σ} - d + 2$

This is a central result in the study of critical phenomena. It shows that the six, seemingly independent, critical exponents are all determined by just two fundamental numbers from the CFT: $Δ_{σ}$ and $Δ_{ϵ}$ . This immediately explains the long-observed scaling relations (e.g., $α + 2 β + γ = 2$ ), which must be true if they are all derived from the same underlying parameters. You can verify this identity using the expressions above.

Test your understanding!

The 2D Ising model is one of the most famous exactly solved models in statistical mechanics. Its critical point is described by a CFT with central charge $c = 1/2$ . The exact scaling dimensions of the spin and energy operators are known to be:

$Δ_{σ} = 1/8$
$Δ_{ϵ} = 1$

Using our dictionary and setting $d = 2$ , calculate the six critical exponents for the 2D Ising model.

Show answer

Let's plug $d = 2$ , $Δ_{σ} = 1/8$ , and $Δ_{ϵ} = 1$ into our formulas:

$ν = \frac{1}{d - Δ _{ϵ}} = \frac{1}{2 - 1} = 1$
$α = 2 - \frac{d}{d - Δ _{ϵ}} = 2 - \frac{2}{2 - 1} = 2 - 2 = 0$ . (A value of 0 implies a logarithmic divergence in the specific heat, $C \sim lo g ∣ t ∣$ , which is the correct result.)
$β = \frac{Δ _{σ}}{d - Δ _{ϵ}} = \frac{1/8}{2 - 1} = 1/8$
$η = 2 Δ_{σ} - d + 2 = 2 (1/8) - 2 + 2 = 1/4$
$γ = \frac{d - 2 Δ _{σ}}{d - Δ _{ϵ}} = \frac{2 - 2 ( 1/8 )}{2 - 1} = 2 - 1/4 = 7/4$
$δ = \frac{d - Δ _{σ}}{Δ _{σ}} = \frac{2 - 1/8}{1/8} = \frac{15/8}{1/8} = 15$

These are the famous, exact critical exponents for the 2D Ising model, derived here from the fundamental data of the corresponding CFT.

For a more detailed derivation of these relations from a pure statistical mechanics and RG perspective, the following video is a useful resource.

Renormalisation group

The video "Renormalisation group" by Jos Thijssen provides a complete, pedagogical derivation of these scaling laws from the principles of coarse-graining. While a full viewing is not necessary, it can clarify the origins of these formulas from the RG perspective.

You can focus on the final part of the video (starting around 40:38), where the critical exponents are systematically derived. The video uses the notation z and v for quantities related to our d-\Delta_\epsilon and \Delta_\sigma. The key point is to see how the scaling properties of the free energy and correlation length under renormalization lead directly to expressions for the exponents.

Conclusion

Today, we have translated the language of statistical mechanics into the language of Conformal Field Theory. We've seen that the critical exponents describing the universal behavior of a system at a phase transition are not arbitrary numbers but are direct consequences of the scaling dimensions of operators in the CFT that governs the critical point.

Key Takeaways:

Dictionary: We have established a direct mapping between the six main critical exponents ( $α, β, γ, δ, ν, η$ ) and the scaling dimensions of the two most relevant scalar operators ( $Δ_{σ}, Δ_{ϵ}$ ) in the associated CFT.
Predictive Power: If we can determine the scaling dimensions of a CFT (for example, using the conformal bootstrap methods we've studied), we can predict all the critical exponents for any physical system in that universality class.
Universality Explained: The fact that all exponents are determined by a small set of CFT data ( $Δ_{i}$ ) provides a deep explanation for why different microscopic systems have the same critical exponents—they simply flow to the same CFT at long distances.

In our next lesson, we will delve into the mechanism that governs this "flow" of theories: the Renormalization Group (RG). We will formalize the idea of perturbing a CFT and see how operators drive the system towards or away from a critical fixed point, providing the dynamical underpinning for the dictionary we've built today.

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