Novel Defect Universality Classes from Interacting RG Interfaces

TL;DR

This work introduces defect conformal structures arising from localized interactions on an RG interface between two distinct multiscalar CFTs in $d=4-\varepsilon$. By deriving the one-loop beta functions for cubic-interface, line-defect, and surface-defect sectors, it shows that the defect flows depend on the average of the two bulks, generating a rich landscape of fixed points beyond those of a single bulk. The authors compute concrete CFT data for the cubic interface, including nonzero one-point functions, a detailed two-point function for $\phi^2$, and the leading interface free energy, illustrating how the interface modifies conformal data and monotonicity properties. The results provide a framework to classify and explore defect universality under RG interfaces and motivate non-perturbative checks via conformal bootstrap, with potential extensions to more general bulk theories and higher dimensions.

Abstract

We search for new defect universality classes by considering localised interactions placed on an RG interface separating two interacting multiscalar CFTs in $4-\varepsilon$ dimensions. Studying interactions spread throughout the entire interface as well as defects restricted to lines and surfaces within the interface, we find that this setup leads to a great number of additional physical fixed points in the space of conformal defects. At one loop it is possible to interpret these fixed points as coming from defects placed within a single bulk whose interaction is an average of the two sides. This averaging means that it is possible to identify conformal defects with considerably less global symmetry than was possible beforehand. We finally compute conformal data for this setup, and find the free energy associated with these RG interfaces.

Figures (14)

• Figure 1: Fixed points of \ref{['eq:betaone']}, with the choice of bulk tensor \ref{['eq:lambdachoiceOFint']}. The dotted lines indicate the trajectories of solutions for arbitrary $\lambda$. The red circles indicate the physical fixed points in an $O(2)$ bulk, with $\lambda=1/10$, while the green square indicates the trivial defect solution, which is the only fixed point in both the Free bulk and $O(2)$--Free interface. Examining the trajectories, one sees that two of the $O(2)$ fixed points merge at $\lambda=1/12$ and move into the complex plane before they can produce a non-trivial fixed point in the $O(2)$--Free interface.
• Figure 2: Fixed points of \ref{['eq:betaone']} in the $O(2)$--$I\times I$ system. The dotted lines indicate the families of solutions parametrised by $\alpha$ with the choice of bulk tensor \ref{['eq:lambdachoiceOB2int']}. One sees that the two non-trivial fixed points in the $I\times I$ bulk both lead, in the $\alpha\rightarrow 1$ limit, to one of the three $O(2)$ fixed points. the other two $O(2)$ fixed points instead have their origin in the region of strong coupling.
• Figure 3: Fixed points found for an interface with cubic couplings. The values for the $O(3)$ and Cubic bulks were taken from Harribey:2024gjn, and can equivalently be computed using \ref{['eq:betaone']} by setting $\lambda^1_{ijkl}=\lambda^2_{ijkl}$. Note that not shown is the single fixed point for a free bulk, which is located at (72.5147, 6.1402).
• Figure 4: The two diagrams contributing to $\langle\phi_i(x)\rangle$ at $O(\varepsilon)$. The vertical dashed line represent the interface localised at $(\mathbf{x},0)$. We will make the choice that the insertion is on the left side of the interface, but this will not change the counterterms necessary for renormalisation.
• Figure 5: Family of solutions to the line defect beta function \ref{['eq:linebetanormal']} for $O(2)$ symmetric values of $\lambda_{ijkl}$. The values of the coupling for the $O(2)$ bulk ($\alpha=\tfrac{1}{10}$), and the $O(2)$--Free interface ($\alpha=\tfrac{1}{20}$) are shown as the red circle and blue triangle respectively. One sees that as $\alpha$ decreases the fixed point moves towards strong coupling, explaining its non-existence in the free theory.