Conformal Bootstrap in Physics

• Conformal bootstrap is a nonperturbative method that leverages conformal symmetry, the OPE, and crossing symmetry to rigorously constrain the space of conformal field theories.
• The framework translates consistency conditions into tractable numerical or analytical equations, yielding precise predictions for critical exponents and operator dimensions.
• Applications to models like the Yang-Lee edge singularity and the 3D Ising model illustrate its accuracy and significance in both unitary and nonunitary settings.

The conformal bootstrap in physics is a nonperturbative framework that uses the full power of conformal symmetry, the operator product expansion (OPE), and crossing symmetry to rigorously constrain the space of possible conformal field theories (CFTs). By translating these fundamental consistency conditions into tractable mathematical or numerical problems, the conformal bootstrap approach has led to some of the most precise determinations of critical exponents, operator dimensions, and OPE coefficients in statistical and high-energy physics, and has provided a unified language for phenomena ranging from phase transitions in statistical mechanics to holography in quantum gravity.

1. Core Principles: Conformal Symmetry, OPE, and Crossing

At the heart of the conformal bootstrap are three intertwined ingredients:

• Conformal invariance, which dictates the transformation properties of local operators and fixes the form of two- and three-point correlation functions up to constants. For a scalar primary $\phi$ in $D$ dimensions, the four-point function is constrained to the form

$$\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4) \rangle = \frac{g(u,v)}{|x_{12}|^{2\Delta_\phi} |x_{34}|^{2\Delta_\phi}}$$

where $u, v$ are conformal cross-ratios.

• Operator Product Expansion (OPE), which encodes the fusion algebra of fields and allows products of local operators to be expanded as sums over primaries and their descendants:

$$\phi(x_1)\phi(x_2) = \sum_{\mathcal{O}} C_{\phi\phi\mathcal{O}}\, |x_{12}|^{\Delta_{\mathcal{O}} - 2\Delta_\phi}\, [\mathcal{O}(x_2) + \dots]$$

This expansion is exact inside correlation functions and, in conformal theories, converges under general circumstances.

• Crossing symmetry, which follows from the associativity of the OPE and requires equivalence of different OPE channel expansions for multi-point correlators. For four-point functions, crossing translates into a functional equation for $g(u,v)$ given by

$g(u,v) = g(v,u)$

or, after expansion in conformal blocks $G_{\Delta,\ell}(u,v)$ and OPE coefficients $p_{\Delta,\ell}$,

$$g(u,v) = 1 + \sum_{(\Delta,\ell)} p_{\Delta,\ell} G_{\Delta,\ell}(u,v)$$

The intertwining of these features yields rigorous, highly nonlinear constraints on all possible CFT data—scaling dimensions, OPE coefficients, and fusion rules.

2. Numerical Bootstrap Methodologies: Crossing Constraints and Truncations

The modern numerical bootstrap translates the infinite set of crossing symmetry conditions into a finite, computationally tractable set by Taylor expanding the crossing relation about a symmetric point (e.g., $z = \bar z = 1/2$) and working with a truncated subset of operators. The expansion generates one inhomogeneous equation,

$$\sum_{(\Delta,\ell)} p_{(\Delta,\ell)}\, f^{(0,0)}_{\Delta_\phi, \Delta, \ell} = 1$$

and infinitely many homogeneous equations for higher derivatives:

$$\sum_{(\Delta,\ell)} p_{(\Delta,\ell)}\, f^{(2m,n)}_{\Delta_\phi, \Delta, \ell} = 0\,,\quad (m+n \neq 0)$$

where the $f^{(m,n)}$ are derivatives with respect to suitable coordinates.

For a "truncable" CFT, if the expansion is cut off to $N$ conformal blocks, one ends up with $M\ge N$ equations in $N$ unknowns. Nontrivial solutions for the scaling dimensions $\{\Delta_j\}$ and OPE coefficients require the vanishing of all $N \times N$ minors:

$$d_i(\{\Delta\}_N) \equiv \det[f^{(2m_i, n_i)}_{\Delta_\phi, \Delta_j}] = 0$$

This determinant condition produces discrete, finite-dimensional systems whose solutions identify spectra and OPE coefficients consistent with crossing for a given truncated fusion algebra.

3. Applicability to Nonunitary Theories and Minimal Input Requirements

A salient feature of this approach is that unitarity—which provides positivity constraints on OPE coefficients and is necessary for Hilbert space interpretations—is not required: the sole input is knowledge of the fusion rules for low-lying primaries. This allows applications to nonunitary CFTs, such as the Yang-Lee edge singularity, which describes universality classes of critical phenomena not captured by conventional (unitary) quantum field theory.

For a scalar $\phi$ with fusion algebra

$$[\Delta_\phi] \times [\Delta_\phi] = 1 + [\Delta_\phi] + [D,2] + [\Delta_4,4] + \ldots$$

(as is the case for Yang-Lee), the truncated bootstrap fully determines all scaling dimensions and OPE coefficients within the truncation—no adjustable free parameters remain.

4. Explicit Examples: Yang-Lee Edge Singularity and Ising Model

Yang-Lee Edge Singularity

For $D=3$ (three dimensions), by truncating to the identity, $\phi$, the stress tensor $[3,2]$, and a spin-4 operator $[\Delta_4,4]$, and imposing the $3\times3$ minor vanishing, one obtains:

• $\Delta_\phi \simeq 0.213$
• $\Delta_4 \simeq 4.49$
• OPE coefficients $p_{\Delta_\phi}\simeq -3.91$, $p_{3,2}\simeq 0.006$

Applying the Fisher relation for the edge exponent,

$\sigma = \frac{\Delta_\phi}{D - \Delta_\phi}$

gives $\sigma \simeq 0.076$, in close agreement with best numerical estimates. Analogous computations in $D=4$ yield $\Delta_\phi \simeq 0.823$, $\sigma \simeq 0.259$.

3D Critical Ising Model

As a consistency check, the method is applied to the Ising CFT in $D=3$ with a more elaborate fusion algebra (including operators corresponding to $\phi^2$, $\phi^4$, $\phi^6$). Using a $5\times5$ determinant for the truncated spectrum, the bootstrap gives

• $\Delta_{\phi^2} \simeq 1.447$
• $\Delta_\phi \simeq 0.518$

These match precisely to high-precision numerics, e.g., $\Delta_\phi \simeq 0.5182(2)$, illustrating that the low-order truncation captures critical exponents with high accuracy.

5. Theoretical Consequences: Precision and Universality

The close agreement of bootstrap-determined exponents (e.g., $\sigma$ for Yang-Lee: $0.076$ in 3d vs. series result $0.077(2)$; $0.259$ in 4d vs. series $0.258(5)$) with independent numerical and analytic results provides powerful evidence of the method's reliability, even with a small set of primary operators. This demonstrates that the truncated conformal block expansion can reproduce critical properties of strongly coupled, possibly nonunitary models across various universality classes.

This method thus:

• Efficiently implements nonperturbative constraints from crossing symmetry for both unitary and nonunitary CFTs,
• Produces explicit and accurate predictions for scaling dimensions and universal observables with minimal phenomenological input,
• Passes nontrivial consistency checks in paradigmatic systems such as the 3d Ising model.

6. Computational Strategies and Practical Implementation

The procedure consists of:

• Writing the four-point crossing equation in terms of conformal block expansions,
• Taylor expanding around a crossing-symmetric point to generate a system of homogeneous and inhomogeneous equations for the truncated set,
• Expressing these as determinant conditions on the scaling dimensions of the low-lying spectrum,
• Solving for discrete allowed $\{\Delta_j\}$ and extracting OPE coefficients from the corresponding eigenvalues of the truncated system.

The computational requirements scale polynomially in the number of included primaries, enabling efficient solution even in higher dimensions or with broad fusion rules.

7. Limitations, Scope, and Broader Impact

While the method provides nonperturbative predictions for theories where the relevant fusion rules are known or can be hypothesized, its accuracy diminishes if important low-dimension operators are omitted from the truncation. There is also no guarantee that such a truncated solution corresponds to a physical CFT beyond the truncation.

Nevertheless, this approach shows that much of the information about a CFT is controlled by a small set of low-lying data, so long as crossing symmetry and the fusion algebra are enforced. The method has had significant impact in constraining and classifying CFTs in diverse contexts, offering a powerful framework for both theorists and computational practitioners in strongly coupled quantum and statistical field theories.