Boundary and Interface CFTs from the Conformal Bootstrap (1502.07217v4)

Abstract: We explore some consequences of the crossing symmetry for defect conformal field theories, focusing on codimension one defects like flat boundaries or interfaces. We study surface transitions of the 3d Ising and other O(N) models through numerical solutions to the crossing equations with the method of determinants. In the extraordinary transition, where the low-lying spectrum of the surface operators is known, we use the bootstrap equations to obtain information on the bulk spectrum of the theory. In the ordinary transition the knowledge of the low-lying bulk spectrum allows to calculate the scale dimension of the relevant surface operator, which compares well with known results of two-loop calculations in 3d. Estimates of various OPE coefficients are also obtained. We also analyze in 4-epsilon dimensions the renormalization group interface between the O(N) model and the free theory and check numerically the results in 3d.

• The paper leverages the conformal bootstrap approach to extract scaling dimensions of boundary operators in defect CFTs.
• It employs the method of determinants to analyze surface transitions in the 3d Ising and O(N) models with results consistent with perturbative calculations.
• The study introduces an RG domain wall linking UV and IR regimes, offering critical insights into interface-induced phenomena.

Boundary and Interface CFTs from the Conformal Bootstrap: A Summary

The paper presents a comprehensive paper of conformal field theories (CFTs) featuring boundaries and interfaces, employing the conformal bootstrap approach to extract useful information about defect CFTs, particularly focusing on codimension-one defects such as flat boundaries or interfaces. Notably, the research applies the method of determinants to analyze surface transitions in the 3d Ising model and other $O(N)$ models.

Methodology and Findings

The authors leverage the crossing symmetry of defect conformal field theories to explore surface operators and bulk spectra. For the extraordinary transition—a situation where the low-lying spectrum of surface operators is known—the bootstrap method is utilized to gather insights into the bulk spectrum. Conversely, for the ordinary transition, the known low-lying bulk spectrum facilitates the calculation of the scale dimension of interested surface operators, allowing comparison with two-loop calculations in 3d.

In their paper, the researchers decoded several numerical results:

• 3d Ising Model Analysis: The application of the conformal bootstrap method via the method of determinants proved efficacious in determining the scaling dimensions and operator product expansion (OPE) coefficients for boundary operators. For instance, a stable solution was found for the ordinary transition, yielding a scaling dimension that agreed well with two-loop calculations.
• RG Domain Wall: The paper introduces an RG domain wall model, representing an interface connecting the perturbative regime to the fixed-point IR regime in the $O(N)$ model and free theory. The insights derived from weak coupling analysis reveal critical behavior across the interface. Notably, the norm of the displacement operator proved to be a pivotal semantic link between UV and IR theories, highlighting transparency across the interface.
• Comparative Analysis: Across various $O(N)$ models (including $N=0,1,2,3$), the bootstrap results for the ordinary transition were consistent with other computational approaches such as Monte Carlo simulations and perturbation expansions. This uniformity underscores the efficacy and broad applicability of the bootstrap method in assessing critical exponents and scaling dimensions.

The robustness of the determinant method is further supported by its ability to provide predictions and comparative data, such as the scale dimensions of the relevant surface operators, across multiple dimensions and theoretical setups.

Practical and Theoretical Implications

The findings have profound implications for the theoretical landscape of defect CFTs, providing a technically adept framework for examining critical phenomena with nontrivial boundary conditions. The conformal bootstrap, complemented by the determinant method, stands out as a powerful analytical tool for probing defect CFTs. Practically, these methods enrich our understanding of surface transitions and critical behavior in statistical field theories, potentially guiding experimental investigations in condensed matter physics.

In terms of future research avenues, the paper provides a foundation for extending the bootstrap approach to higher dimensions or exploring its implications in other field theories. Moreover, the application of these techniques to both higher-order perturbative regimes and large-$N$ limits can yield further insights into the dynamics and stability of defect interfaces.

Overall, the paper performed in the paper underscores the utility of the conformal bootstrap approach in solving complex boundary and interface problems, paving the way for its application across various fields of theoretical physics and further enriching our understanding of conformal invariance and scale properties.