Bootstrapping the O(N) Vector Models

Abstract: We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N) singlet and symmetric tensor operators appearing in the $\phi_i \times \phi_j$ OPE, where $\phi_i$ is a fundamental of O(N). Comparing these bounds to previous determinations of critical exponents in the O(N) vector models, we find strong numerical evidence that the O(N) vector models saturate the bootstrap constraints at all values of N. We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N.

• The paper demonstrates that operator dimensions in O(N) vector models saturate the bootstrap bounds across all N values.
• It applies semidefinite programming to convert infinite constraints into a numerically tractable analysis via rational approximations of conformal blocks.
• Findings include precise bounds consistent with 1/N expansion and lattice simulations, offering robust predictions for critical phenomena.

Overview of Bootstrapping the $O(N)$ Vector Models

The paper discusses the application of the conformal bootstrap technique to three-dimensional conformal field theories (CFTs) with $O(N)$ symmetry. Specifically, it derives upper bounds for the scaling dimensions of the lowest $O(N)$ singlet and symmetric tensor operators in the operator product expansion (OPE) of scalar fields. By contrasting these bounds with known critical exponents for the $O(N)$ vector models, the authors demonstrate that these models saturate the bootstrap constraints across all values of $N$. They also set lower bounds on the central charge and find that these are consistent with predictions from the $1/N$ expansion.

Methodology

The analysis focuses on scalar operators in 3D CFTs, transforming under the $O(N)$ group. The critical $O(N)$ vector models describing second-order phase transitions, which are also tractable in a $1/N$ expansion, were prime candidates for this study. The primary objective was to place upper bounds on the dimensions of the singlet ($S$) and symmetric tensor ($T_{ij}$) operators. Simultaneously, lower bounds were aimed at determining the central charge $c$.

To efficiently handle these problems, techniques from semidefinite programming were used. This allowed the authors to manage infinite constraints by converting the problem into a numerically feasible form, using rational approximations of conformal blocks over various space-time dimensions.

Key Results and Comparison

The authors found upper bounds on $\Delta_S$ and $\Delta_T$ across various $N$ values, where $\Delta_\phi$ is the dimension of the external scalar. An intriguing outcome is the substantiation of the prediction that operator dimensions in $O(N)$ vector models saturate these numerically derived bounds, effectively marking them as special within crossing symmetric spaces.

Numerically precise determinations for the dimensions were also consistent with those from lattice simulations and theoretical predictions. Additionally, the bootstrap predicted constraints on the central charge aligned with the large $N$ limit to first order in $1/N$, confirming previous results.

Implications and Future Directions

The research has significant implications for both theoretical and experimental physics. For $O(N)$ vector models, the results refine known bounds, possibly leading to updated critical exponents. Such improvements are relevant to condensed matter systems and isotropic magnets wherein $O(N)$ symmetries have experimental realizations. Furthermore, the approach offers potential pathways for exploring higher-spin theories using non-perturbative methods.

This work opens pathways for further generalization to other correlators, including incorporation of additional operators (e.g., spinor and vector exchange). The application of these techniques to eight-dimensional space, as suggested by extending the recursion relation for conformal blocks, could potentially help verify or refute the existence of fixed-point theories within certain ranges of dimensions.

Finally, the bootstrapping method's ability to place constraints without explicitly solving the theory demonstrates its power and promise for exploring complex systems, hence guiding experimental verifications and theoretical explorations alike. Future works may expand or deviate from the assumptions on operator gaps, further refining the specificity of such constraints. This study thus represents a notable contribution to the computational and theoretical mechanics of conformal bootstrap approaches, marking a step towards the non-perturbative understanding of quantum field theories.