Precision Islands in the Ising and $O(N)$ Models

Abstract: We make precise determinations of the leading scaling dimensions and operator product expansion (OPE) coefficients in the 3d Ising, $O(2)$, and $O(3)$ models from the conformal bootstrap with mixed correlators. We improve on previous studies by scanning over possible relative values of the leading OPE coefficients, which incorporates the physical information that there is only a single operator at a given scaling dimension. The scaling dimensions and OPE coefficients obtained for the 3d Ising model, $(\Delta_{\sigma}, \Delta_{\epsilon},\lambda_{\sigma\sigma\epsilon}, \lambda_{\epsilon\epsilon\epsilon}) = (0.5181489(10), 1.412625(10), 1.0518537(41), 1.532435(19))$, give the most precise determinations of these quantities to date.

• The paper achieves unprecedented precision in determining scaling dimensions and OPE coefficients by applying the conformal bootstrap to mixed correlators.
• It employs exclusion plots to define a three-dimensional 'island' in parameter space, refining benchmarks for the 3D Ising, O(2), and O(3) models.
• Key results such as Δσ = 0.5181489 and Δε = 1.412625 for the 3D Ising model offer critical benchmarks for both simulations and experimental validations.

Precision Islands in the Ising and $O(N)$ Models: A Comprehensive Overview

In the pursuit of understanding conformal field theories in dimensions greater than two, the conformal bootstrap method has proven to be an indispensable tool, providing numerical precision previously unattainable with traditional methods. The paper "Precision Islands in the Ising and $O(N)$ Models" by Kos et al. builds upon previous studies in this area, notably improving precision in the calculation of scaling dimensions and operator product expansion (OPE) coefficients within these models. The Ising model, a critical lattice model in three dimensions, serves as the focal point for much of the study, with extensions to the $O(2)$ and $O(3)$ models marking further contributions.

Numerical Precision via the Conformal Bootstrap

The authors employ the conformal bootstrap to analyze mixed correlators, allowing them to make precision determinations of leading scaling dimensions and OPE coefficients in the 3D Ising, $O(2)$, and $O(3)$ models. A key advancement in this work is the consideration of exclusion plots in the space of both OPE coefficients and dimensions, leading to the identification of a three-dimensional parameter space, colloquially termed an "island," wherein the scaling dimensions and relative values of leading OPE coefficients lie. This approach is significant because it circumvents the practical limitations posed by degeneracies in the conformal field theory spectrum, offering a more constrained and hence precise determination of the theory's parameters.

Results and Interpretation

For the 3D Ising model, the scaling dimensions $(\Delta_{\sigma}, \Delta_{\epsilon})$ are determined with impressive precision at $\Delta_{\sigma} = 0.5181489(10)$ and $\Delta_{\epsilon} = 1.412625(10)$. Furthermore, the ratio of OPE coefficients $$\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon} = 1.456889(50)$$ was calculated, allowing determination of the coefficients themselves: $\lambda_{\sigma\sigma\epsilon} = 1.0518537(41)$ and $\lambda_{\epsilon\epsilon\epsilon} = 1.532435(19)$. These results are not only significant in their precision but also serve as crucial benchmarks for comparisons against Monte Carlo simulations and experiments, especially in the case of determining critical exponents.

For the $O(2)$ and $O(3)$ models, refinement continues, albeit with precision still lagging behind that of the Ising model. Nonetheless, the results obtained for these models show alignment with previous Monte Carlo determinations while hinting at discrepancies in experimental data analysis, specifically in the $O(2)$ model evidenced by tension between experimental determination of critical exponents and the results predicted by the bootstrap.

Implications and Future Directions

The implications of these findings are twofold: first, the increased precision in critical parameters marks a substantial step forward in non-perturbative quantum field theory calculations, offering deeper insights into the universality classes of statistical models at criticality. Second, the methods employed can be extended to investigate other theories where conformal field theories play a crucial role, such as in the study of quantum gravity or physics beyond the Standard Model.

Looking toward the future, the methodology presented—leveraging exclusion plots and refined numerical techniques in the conformal bootstrap—holds promise for even more efficient isolation of CFT spectra, potentially offering solutions to long-standing problems such as determining the conformal windows of three-dimensional quantum electrodynamics (QED) and four-dimensional quantum chromodynamics (QCD).

Conclusion

"Precision Islands in the Ising and $O(N)$ Models" exemplifies the growing power of the conformal bootstrap method in quantitative theoretical physics. By achieving unprecedented precision in parameters of widely studied models, Kos et al. mark a critical step towards a predictive and comprehensive framework for non-perturbative conformal field theories, heralding a new era in the study of critical phenomena and beyond.