The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT (1612.08471v1)

Abstract: We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2,R) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of several infinite families of operators in terms of the initial data ${\Delta_\sigma,\Delta_\epsilon,f_{\sigma\sigma\epsilon},f_{\epsilon\epsilon\epsilon},c_T}$. The analytic results agree with numerics to high precision for about 100 low-twist operators (correctly accounting for O(1) mixing effects between large-spin families). Plugging these results back into the crossing equations, we obtain approximate analytic constraints on the initial data.

• The paper introduces novel lightcone bootstrap methods to compute infinite sums of conformal blocks and extract key operator data.
• The study analytically approximates operator dimensions and OPE coefficients for nearly 100 low-twist operators with high precision.
• The computational advances bridge numerical and analytical CFT methods, offering a robust template for future theoretical explorations.

The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT

This paper presents an in-depth computational investigation into the spectrum of the three-dimensional Ising Conformal Field Theory (CFT) using the lightcone bootstrap approach. The Ising model serves as a foundational system, encapsulating crucial principles for the investigation of phase transitions and critical phenomena. A specific focus is placed on computational techniques to extract operator dimensions and Operator Product Expansion (OPE) coefficients, extending both numerical and analytical methods.

Computational Techniques for Infinite Sums of Conformal Blocks

The paper introduces novel techniques for computing infinite sums of $\SL(2, \R)$ conformal blocks, providing a systematic approach to handle series expansions and asymptotic behaviors in the context of large-spin expansions. These methodologies are critical for reverse-engineering solutions to crossing symmetry, especially post numerical spectrum initialization through the bootstrap approach.

Analytical Approximation of Operator Dimensions and OPE Coefficients

By employing these computational tools, the paper succeeds in analytically approximating operator dimensions and OPE coefficients for several infinite families of operators using known initial data $\{\De_\sigma, \De_\epsilon, f_{\sigma\sigma\epsilon}, f_{\epsilon\epsilon\epsilon}, c_T\}$. Surprisingly, the approximation matches numerical results with high precision for approximately 100 low-twist operators. This is facilitated by correctly accounting for mixing effects occurring between large-spin families, a significant computational achievement highlighted in this research.

Solving the Lightcone Bootstrap

The solution to the lightcone bootstrap to all orders in an asymptotic expansion in large spin, as proposed in this work, represents a crucial analytical step forward. This advances the goal of theoretically deriving correlator information traditionally reliant on heavy numerical computation. Notably, this involves extending the applicability of the large-spin expansion techniques to spans where operator twists converge to known limits.

Challenges and Computational Efficacy

A noteworthy challenge tackled by the authors is the effective separation and treatment of Casimir-singular and Casimir-regular terms in infinite series. By leveraging customized asymptotic expansions, they are able to efficiently derive key insights into operator behavior, thereby informing constraints on theoretical initial data through the approximate analytic constraints imposed by crossing equations.

Implications for Future Research

Theoretically speaking, this rigorous computation paves the way for enhanced understanding and application of conformal symmetry principles, extending beyond the Ising model to broader classes of CFTs. Practically, the paper provides a template for employing combined numerical and analytical methodologies to derive critical properties of higher-dimensional field theories, bearing significance for both theoretical physics and applied statistical mechanics.

Concluding Analysis

In conclusion, by enhancing computational methods for conformal block decomposition, the researchers have taken a substantial step in closing the gap between numerical and analytical CFT studies. Their scholarly contribution reverberates as an example of computational refinement and theoretical vigor, charting a path forward for CFT exploration and symmetry analysis broadly in complex physical systems.