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Solving the 3D Ising Model with the Conformal Bootstrap (1203.6064v3)

Published 27 Mar 2012 in hep-th and cond-mat.stat-mech

Abstract: We study the constraints of crossing symmetry and unitarity in general 3D Conformal Field Theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and OPE coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Citations (877)

Summary

  • The paper demonstrates that the conformal bootstrap effectively determines operator dimensions in the 3D Ising model.
  • It provides numerical bounds on scalar operators that align with known critical exponents from theory and experiments.
  • The methodology opens avenues for exploring higher spin operators and holographic dualities in conformal field theories.

Solving the 3D Ising Model with the Conformal Bootstrap

The paper, "Solving the 3D Ising Model with the Conformal Bootstrap," presents an in-depth paper of the constraints inherent in crossing symmetry and unitarity within the framework of three-dimensional conformal field theories (CFTs). Specifically, the research addresses the 3D Ising model at its critical temperature, a problem of considerable interest due to its connections with real-world phase transitions.

Overview of the Methodology

The authors employ the conformal bootstrap approach, a non-perturbative technique leveraging the symmetry properties of conformal field theories. The focus is on understanding the operator dimensions and their corresponding operator product expansion (OPE) coefficients by solving crossing symmetry equations for four-point correlation functions.

The researchers derive explicit results for conformal blocks in the context of four-point functions of scalar fields and introduce an efficient methodology for computing these blocks for arbitrary spacetime dimensions. This enables a systematic paper of the bounds on operator dimensions in 3D CFTs. One of the standout findings is that the 3D Ising model occupies a distinct corner point on the boundary of the allowed parameter space in such theories, indicating a unique configuration of operator dimensions characterized by the conformal bootstrap.

Key Findings and Implications

  • Numerical Results: The paper provides strong numerical support for the conjecture that the 3D Ising model's critical behavior can be captured by a CFT with minimal additional symmetries apart from conformal and Z2\mathbf{Z}_2 symmetry. Numerical bounds on dimensions of scalar operators such as Δϵ\Delta_{\epsilon} and Δσ\Delta_{\sigma} are consistent with the known critical exponents derived from other theoretical and experimental methods.
  • Bounds on Operator Dimensions: The examination of scalar operators and their associated dimensions uncover that the 3D Ising model may lie on a boundary of the CFT parameter space, suggesting an underlying significance in its conformal data that distinguishes it from other CFTs.
  • Higher Spin Operators: The research also explores the implications of bounds on the dimensions of higher spin operators, which have important consequences for theories characterized by weakly broken higher spin symmetries, bridging connections with holographic duals in higher-dimensional spaces.

Speculation on Future Developments

The work highlights potential pathways for future research, including:

  • Enhanced Bootstrap Approaches: Building upon this foundational research, subsequent studies could explore the incorporation of additional operators (including higher spin) in the bootstrap constraints to refine and potentially constrain entire swathes of the unseen 3D CFT landscape.
  • Connection to Holography: Theoretical development in the context of the AdS/CFT correspondence may gain insights from these bootstrap techniques, particularly in higher spin theories and their putative holographic descriptions.
  • Exploration of Other Universality Classes: The methods developed could be applied to other critical models in differing dimensions, potentially shedding light on universality classes outside the scope of the Ising model.

Conclusion

This paper provides important advancements in the application of the conformal bootstrap to 3D CFTs through a rigorous computational approach. Its implications stretch across theoretical physics, not only broadening understanding of the 3D Ising model but also offering a template for future exploration of conformal field theories and their diverse applications amidst various physical phenomena and mathematical structures.

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