Numerical Conformal Bootstrap

• Numerical conformal bootstrap is a computational method that enforces crossing symmetry to rigorously bound operator dimensions and OPE coefficients in conformal field theories.
• It employs Taylor expansion, truncation strategies, and determinantal equations to convert infinite crossing constraints into a finite system, enabling precise numerical analysis.
• The approach yields accurate estimates of critical exponents in models like the Yang-Lee edge singularity and 3d Ising model, applicable to both unitary and nonunitary cases.

The numerical conformal bootstrap is a computational methodology for extracting rigorous, nonperturbative constraints on the data of conformal field theories (CFTs)—specifically, the spectra of operator dimensions and operator product expansion (OPE) coefficients—by enforcing crossing symmetry and, when appropriate, unitarity. This approach has become a central tool for analyzing strongly coupled CFTs in various spacetime dimensions and has led to sharp bounds and, in some cases, precision determinations of critical exponents relevant for statistical and quantum phase transitions.

1. Foundations: Crossing Symmetry and Bootstrap Equations

The bootstrap program exploits the constraints imposed by crossing symmetry on four-point functions. For identical scalar operators $\phi$ in $D$ dimensions, the correlator is written as

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

where $u$ and $v$ are conformally invariant cross-ratios. The function $g(u,v)$ is decomposed into a sum over conformal blocks $G_{\Delta, L}(u, v)$, representing the contributions of primary operators of scaling dimension $\Delta$ and spin $L$: $$g(u, v) = 1 + \sum_{\Delta, L} p_{\Delta, L} G_{\Delta, L}(u, v).$$ Crossing symmetry (from permutations of the insertion points) leads to the functional constraint

$$\sum_{\Delta,L} p_{\Delta, L} \frac{v^{\Delta_\phi} G_{\Delta,L}(u, v) - u^{\Delta_\phi} G_{\Delta,L}(v, u)}{u^{\Delta_\phi} - v^{\Delta_\phi}} = 1.$$

These equations enforce a highly nontrivial consistency requirement: any valid CFT spectrum and OPE coefficients must produce a four-point function invariant under channel permutations.

2. Numerical Methodologies: Derivative and Functional Bases

To systematically extract information, the infinite set of bootstrap equations is truncated and recast as a finite or semi-infinite system suitable for numerical analysis.

• Taylor/Derivative Expansion: Crossing equations are expanded in Taylor series around a symmetric point (e.g., $z = \bar{z} = 1/2$ or $a=1, b=0$ under a change of variables), yielding a set of conditions on derivatives $f^{(m,n)}_{\Delta_\phi, \Delta, L}$:

$$f^{(m,n)}_{\alpha, \beta} = \partial_a^m \partial_b^n \left[ \frac{v^\alpha G_\beta(u,v) - u^\alpha G_\beta(v,u)}{u^\alpha-v^\alpha} \right]_{a=1, b=0}.$$

• Homogeneous and Inhomogeneous Constraints: The Taylor expansion leads to a single inhomogeneous constraint (from derivatives of order zero) and an infinite set of homogeneous constraints (from higher derivatives).
• Truncation and Determinantal Equations: Truncating the spectrum to a finite number of low-lying operators leads to a finite linear system for the OPE data. Consistency requires the vanishing of all $n \times n$ minors of the resulting matrix of derivatives, yielding transcendental equations for the allowed $\{\Delta_i\}$:

$$d_i(\{\Delta\}_n) \equiv \det \left[ f^{(2m_i, n_i)}_{\Delta_\phi, \Delta_j} \right] = 0, \quad i=1, \ldots, \kappa.$$

This is conceptually analogous to the Vafa equations for 2d rational CFTs but generalized to higher-dimensional and non-unitary settings via transcendental (rather than Diophantine) constraints.

• Functional/Analytic Bases: Recent advances employ analytic functionals, often constructed to have zeros at the generalized free spectrum, yielding much faster convergence and more robust handling of high-dimension or large-spin tails (Paulos et al., 2019, Ghosh et al., 2023).

3. Applicability: Unitary and Nonunitary, Truncable Theories

Classical numerical bootstrap strategies predominantly focused on unitary CFTs—where unitarity enforces positive-definiteness of the OPE coefficients squared. The methodology outlined in (Gliozzi, 2013) demonstrates that:

• Given knowledge of the fusion algebra (the set of operator product rules for low-lying primaries of the form $$[\Delta_\phi] \times [\Delta_\phi] = \sum_i N_i [\Delta_i, L_i]$$),
• Even for nonunitary or more general truncable CFTs (where the four-point function is well-approximated by a finite number of conformal blocks), the same approach applies.
• There are then no free parameters in the equations: if the fusion algebra is known, the scaling dimensions and OPE coefficients are completely determined, provided a consistent, finite solution exists.

This extension to nonunitary systems allows, for instance, the paper of models such as the Yang-Lee edge singularity (described by a $\phi^3$ interaction with imaginary coupling) and other critical points that evade standard positivity-based methods.

4. Case Studies and Numerical Results

• Yang-Lee Edge Singularity: Implementing the method up to spin-4 and solving for the common zeros of selected $3 \times 3$ minors yields:
  • In $d=3$: $\Delta_\phi \simeq 0.213$, $\Delta_4 \simeq 4.49$, and $p_{\Delta_\phi} \simeq -3.91$; the derived edge exponent $\sigma = \Delta_\phi/(D-\Delta_\phi) \simeq 0.076$, matching well the best numerical estimates $\sigma = 0.077(2)$.
  • In $d=4$: $\Delta_\phi \simeq 0.823$, $\Delta_4 \simeq 5.71$, $\sigma \simeq 0.259$, again compared favorably to $\sigma = 0.258(5)$.
• 3d Critical Ising Model: Including additional low-lying even-scalar operators (e.g., associated with $\phi^4$ and $\phi^6$ fields), a $5 \times 5$ minor gives a nonlinear constraint $F(\Delta_\phi, \Delta_{\phi^2}) = 0$. Solution yields $\Delta_{\phi^2} \simeq 1.447$ and reinserting this value gives $\Delta_\phi \simeq 0.518$, in excellent agreement with Monte Carlo and numerical results ($\Delta_\phi \simeq 0.5182$, $\Delta_{\phi^2} \simeq 1.4130$).
• Consistency and Convergence: The accuracy improves systematically by increasing the truncation level (i.e., the number of conformal blocks included), and the method is benchmarked by comparison to exact values in $d=2$ (for which minimal models provide analytical solutions).

5. Implications, Scope, and Limitations

• This framework extends the regime of numerical bootstrap from the "boundary" of the allowed parameter space (the unitarity saturated edge) into the interior, provided sufficient information about the fusion rules is available.
• The approach is not tied to positivity or unitarity, enabling systematic analyses of nonunitary CFTs and even of "truncable" models, where the four-point dynamics are dominated by a finite set of low-dimension operators.
• The algebraic form (determinantal vanishing) favors settings where the fusion algebra is sparse and well-defined. In models with a denser or more ambiguous fusion algebra, the practical efficacy may be limited by the need to truncate at higher operator levels.
• The similarity to the 2d Vafa equations highlights the unifying structural features across dimensions and the potential for classifying large classes of CFTs through transcendental analogues of known Diophantine solutions.

6. Comparison to Other Bootstrap Strategies

• In the "method of minors" or determinantal approach, emphasis is on encoding crossing symmetry as a nonlinear algebraic system, rather than enforcing positivity across the spectrum as in positivity-based semidefinite programming (SDP) or linear programming (LP) methods.
• When the fusion algebra is not fully known or is infinite, general techniques based on convex programming, such as those implemented in semidefinite programming solvers (e.g., SDPB), remain essential.
• The approach in (Gliozzi, 2013) complements and sometimes surpasses unitarity-based strategies in situations where nonunitarity or truncability are central, as in the Yang–Lee criticality.
• A plausible implication is that this algebraic bootstrap could serve as a classification tool for CFTs with few low-lying primaries, while traditional SDP/LP approaches remain superior for unitary or more generic large-spectrum problems.

7. Prospects and Conceptual Significance

The reformulation of the numerical conformal bootstrap as a system of transcendental equations derived from finite truncations of the crossing relations and fueled by knowledge of the fusion algebra significantly broadens the scope of the bootstrap program. This suggests an avenue toward an algebraic classification of CFTs beyond the minimal model paradigm and enables precision studies of critical exponents in nonunitary systems, as demonstrated by agreement with most recent numerical and Monte Carlo exponents.

The systematic Taylor expansion approach, construction of vanishing minors, and associated transcendental constraints together constitute a powerful and efficient alternative to traditional positivity-based bootstrap algorithms, especially valuable in the paper and potential classification of nonunitary or sparsely populated CFT spectra.