Bootstrapping Mixed Correlators in the 3D Ising Model (1406.4858v1)

Abstract: We study the conformal bootstrap for systems of correlators involving non-identical operators. The constraints of crossing symmetry and unitarity for such mixed correlators can be phrased in the language of semidefinite programming. We apply this formalism to the simplest system of mixed correlators in 3D CFTs with a $\mathbb{Z}2$ global symmetry. For the leading $\mathbb{Z}_2$-odd operator $\sigma$ and $\mathbb{Z}_2$-even operator $\epsilon$, we obtain numerical constraints on the allowed dimensions $(\Delta\sigma, \Delta_\epsilon)$ assuming that $\sigma$ and $\epsilon$ are the only relevant scalars in the theory. These constraints yield a small closed region in $(\Delta_\sigma, \Delta_\epsilon)$ space compatible with the known values in the 3D Ising CFT.

• The paper introduces a novel method that transforms quadratic crossing symmetry conditions into semidefinite programming problems.
• It obtains precise numerical bounds on operator dimensions for the 3D Ising model that closely match Monte Carlo and analytical results.
• The approach enhances conformal bootstrap techniques, offering a framework to explore and classify a wider range of 3D conformal field theories.

Insights from "Bootstrapping Mixed Correlators in the 3D Ising Model"

The paper "Bootstrapping Mixed Correlators in the 3D Ising Model" by Filip Kos, David Poland, and David Simmons-Duffin stands as a notable contribution to the conformal bootstrap program, specifically focusing on constraints provided by mixed correlators in three-dimensional conformal field theories (CFTs) with a $\mathbb{Z}_2$ symmetry. The work advances the numerical bootstrap technique by extending it to systems of four-point functions involving non-identical operators, thereby broadening the analytical reach beyond that of single-correlator studies.

The Theoretical Framework and Approach

The paper addresses the need to explore mixed correlators—correlators that involve different primary operators—within the conformal bootstrap framework. Traditional studies have predominantly focused on single-type operators, where sums over conformal blocks are utilized to generate numerical constraints. However, in mixed correlator scenarios, quantum fields with non-identical operators invoke more complex crossing symmetry and unitarity requirements. These become inherently quadratic rather than merely linear, as is the case with identical operators.

The paper employs semidefinite programming (SDP) to handle this complexity, transforming the quadratic constraints into SDP conditions. The authors resolve the mathematical and computational challenges associated with solving these programs, particularly for mixed correlators, by redefining them into SDP problems with reduced variables. This methodology allows constraints to be handled efficiently via computational models.

Numerical Results and Significance

Focusing on the three-dimensional Ising model—a system renowned for its role in statistical and quantum physics—the paper investigates mixed four-point functions with leading $\mathbb{Z}_2$-odd and $\mathbb{Z}_2$-even operators. It derives compelling numerical bounds on operator dimensions $(\Delta_\sigma, \Delta_\epsilon)$, assuming that these operators are the sole relevant scalars in the theory.

One of the strong points of the research is the numerical pinpointing of a feasible and narrow region for $(\Delta_\sigma, \Delta_\epsilon)$ around the known values, demonstrating striking agreement with Monte Carlo simulations and prior analytical conjectures. The results affirm the potency of the mixed correlator bootstrap in not only verifying known predictions but perhaps ultimately aiding the classification of other isolated CFTs in three dimensions.

Additionally, upper bounds on the dimension of the second-lowest $\mathbb{Z}_2$-odd scalar $\sigma'$, give insight into the potential structure of the spin spectrum in the model. These results may have broader implications for any $\mathbb{Z}_2$-symmetric 3D CFT, suggesting that constraints are sufficient potentially to distinguish the Ising CFT as a unique feature of operator gaps.

Implications and Future Directions

The implications of this work are both profound and practical. The research demonstrates that mixed correlators can yield more stringent constraints than single correlators when mild assumptions about spectrum gaps are introduced. This possibly marks a precursor to refinement in bootstrap methodologies, opening avenues for rigorous exploration of the space of CFTs beyond those with known degrees of freedom or symmetries.

Moreover, the paper significantly advances computational techniques relevant to the field of theoretical physics. By demonstrating the scalability and utility of semidefinite programming for solving these types of ScFTs (conformal field theories), researchers are afforded a robust toolset to tackle higher-dimensional problems or those involving intricately coupled operators.

In future iterations, scaling up this approach to include larger classes of operators, higher spin states, and extending it to incorporate global symmetries are anticipated. These efforts could potentially lead to a deeper understanding of the non-perturbative structure of field theories, enhancing the predictability and precision of critical phenomena predictions.

In conclusion, Kos, Poland, and Simmons-Duffin have provided a new lens through which the rich structure of the 3D Ising model can be further probed and understood, potentially leading to isolating other critical models with distinct spectral properties in the field of quantum field theory.