Boundary and Interface CFTs from the Conformal Bootstrap

TL;DR

This work applies the conformal bootstrap, via the determinant/truncation method, to defect CFTs with codimension-one boundaries and interfaces, focusing on surface transitions in the 3d Ising and O(N) models. By solving the crossing equations for bulk two-point functions near a defect and exploiting the folding trick, the authors extract spectra and OPE data for the ordinary, extraordinary, and special boundary transitions, validating results against perturbative and Monte Carlo benchmarks and Ward identities. They also extend the analysis to RG domain walls between the O(N) model and the free theory using ε-expansion, deriving leading mixing patterns between UV and IR primaries and confirming cross-channel consistency with 3d Ising interface data. The paper demonstrates the determinant bootstrap as a viable, cross-checked nonperturbative tool for defect CFTs, providing concrete spectra and OPE coefficients and guiding future explorations of more complex defect configurations.

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.

Figures (9)

• Figure 1: Top panel: paired histograms of the solutions of two different truncations of the crossing equations for the ordinary transition of the 2d Ising model. Left: histogram for the scale dimensions of the first boundary operator in the (2,1,0) truncation. The exact result is at $\widehat{\Delta}=\frac{1}{2}$. Right: the corresponding histogram for the (4,3,0) truncation. Bottom panel: a more detailed view of the latter histogram.
• Figure 2: The left-hand-side of the sum rule (\ref{['eq:crosy']}) for various truncations $(n_{bulk},n_{bdy},0)$ of the two-point function of the 2d Ising model in the ordinary transition. Only in the $n_{bulk}\to\infty$, $n_{bdy}\to\infty$ limit the sum rule is saturated.
• Figure 3: Plot of the 10 $3\times3$ minors made with the first 5 derivatives of the conformal blocks associated with $\varepsilon$, $\varepsilon'$ and $\widehat{O}$ as functions of $\Delta_{\widehat{O}}$. They all vanish approximately at he same point, selecting the allowed value of $\Delta_{\widehat{O}}$.
• Figure 4: Parametric plot of the scaling dimensions of $\Delta_{\varepsilon"}$ and $\Delta_{\varepsilon"'}$ generated by the unknown parameter $\widehat{\Delta}$ in the (4,2,1) truncation. Here we see the effect of the statistical errors on the input data, namely $\Delta_\sigma$, $\Delta_{\varepsilon}$ and $\Delta_{\varepsilon'}$ as well as the effect of the spread of the solutions. Some of these data are presented in table \ref{['tab:3']}.
• Figure 5: Plot of the zeros of some $5\times5$ determinants associated with the fusion rules (\ref{['sigmaepsilonBulk']}) and (\ref{['sigmaepsilonBdy']}).