Bootstrapping the O(N) Archipelago (1504.07997v2)

Abstract: We study 3d CFTs with an $O(N)$ global symmetry using the conformal bootstrap for a system of mixed correlators. Specifically, we consider all nonvanishing scalar four-point functions containing the lowest dimension $O(N)$ vector $\phi_i$ and the lowest dimension $O(N)$ singlet $s$, assumed to be the only relevant operators in their symmetry representations. The constraints of crossing symmetry and unitarity for these four-point functions force the scaling dimensions $(\Delta_\phi, \Delta_s)$ to lie inside small islands. We also make rigorous determinations of current two-point functions in the $O(2)$ and $O(3)$ models, with applications to transport in condensed matter systems.

• The paper employs a conformal bootstrap approach with semidefinite programming to isolate narrow islands for operator scaling dimensions in 3d O(N) models.
• The paper provides precise estimates for the current central charge in O(2) and O(3) models that align with established numerical benchmarks.
• The paper suggests that further refinement of the bootstrap technique could extend to other strongly interacting theories, enhancing both theoretical predictions and experimental validation.

Insights into Bootstrapping the $O(N)$ Archipelago

The paper under consideration provides a rigorous analysis of three-dimensional conformal field theories (3d CFTs) exhibiting $O(N)$ global symmetry through the application of the conformal bootstrap method. The authors aim to explore the constraints put forward by crossing symmetry and unitarity on mixed four-point correlators, consisting of the lowest dimension $O(N)$ vector $\phi_i$ and the lowest dimension singlet $s$. They examine how these constraints can narrow the scaling dimensions $\Delta_\phi$ and $\Delta_s$ to within small regions, referred to as "islands."

Key Findings and Methodology

The researchers employ semidefinite programming techniques to enforce the crossing symmetry of multiple correlators, an approach that goes beyond the standard single-correlator bootstrap analysis. By considering all non-vanishing scalar four-point functions, the paper provides a robust isolation of scaling dimensions for these scalar operators with a significant focus on $N=2, 3, 4$ systems, due to their physical implications and the failure of the large-$N$ expansion in these regimes. Moreover, the analytical framework is not limited to providing dimensional bounds; it also yields precise estimates for the current central charge $C_J$ in $O(2)$ and $O(3)$ models, a parameter pertinent to transport properties in condensed matter systems.

The bootstrapping technique used rigorously tests whether hypothetical spectra of scaling dimensions can be consistent with the assumed symmetries. This is achieved through a systematic examination of the crossing equations for these symmetry-constrained correlators, starting from the framework detailed in previous bootstrap studies and enhanced by modifications that incorporate semidefinite programming.

Numerical Results and Implications

One of the significant outcomes is the determination of narrow islands in parameter space for the operator dimensions $\Delta_\phi$ and $\Delta_s$ in the $O(N)$ vector models. For instance, for $N=2$, commonly related to phenomena like the superfluid transition in ${}^4$He, the research successfully derives bounds that closely encircle known values from Monte Carlo and high-temperature expansion results. However, the paper emphasizes that while these regions do not yet surpass the precision of existing numerical results, they provide notable confirmation and, importantly, point the way toward heightened precision in future studies.

Additionally, the analysis of the current central charge shows that the bootstrap method can quantitatively constrain parameters associated with dynamic responses like conductivity at criticality. This finding presents exciting opportunities for using bootstrap techniques to comprehend physical phenomena beyond theoretical bounds, potentially impacting experimental verifications.

Speculations and Future Prospects

This work opens many avenues for future research. It suggests that further refinement of the bootstrap implementation, especially by exploring additional representations (such as symmetric tensors or higher spin sectors), could move closer to pinpointing the $O(N)$ model data. Moreover, this methodology might be applicable to other strongly interacting theories like Gross-Neveu models or gauge theories, offering insights into their critical properties.

The paper also highlights the profound potential of bootstrapping in tackling operators' spectra in quantum and conformally invariant field theories, incrementally bridging the gap between theoretical expectation and observed physical reality through improved numerical algorithms and extended datasets.

While the current conclusions are robust under existing constraints, the paper envisions a future where bootstrap techniques continually evolve, both in breadth and depth, sculpting a clearer understanding of CFT landscapes, essentially enhancing theoretical predictions and stipulating future observational studies in high-energy and condensed matter physics.