Spin Impurities, Wilson Lines and Semiclassics

TL;DR

The paper develops a comprehensive framework for line defects with large quantum numbers in conformal field theories. It constructs a semiclassical, double-scaling approach for spin impurities, both in a free $O(3)$ bulk and in the interacting Wilson-Fisher $O(3)$ model, revealing a decoupled $S^2$ sector and a close relation to the pinning-field DCFT for large spin, with concrete predictions for operator dimensions and defect g-functions. It then turns to large-representation 1/2-BPS Wilson lines in rank-1 $\mathcal{N}=2$ SCFTs, where localization reduces observables to 1D integrals and the large-$s$ limit maps to the Coulomb-branch EFT, yielding universal formulas involving the $a$-anomaly and the $g$-function, and connecting to non-Lagrangian theories. The work provides precise, testable predictions for three-dimensional magnets and magnet-like impurities, and establishes a unifying picture in which large quantum numbers simplify defect dynamics, enabling controlled perturbative and semiclassical analyses across diverse DCFT settings.

Abstract

We consider line defects with large quantum numbers in conformal field theories. First, we consider spin impurities, both for a free scalar triplet and in the Wilson-Fisher $O(3)$ model. For the free scalar triplet, we find a rich phase diagram that includes a perturbative fixed point, a new nonperturbative fixed point, and runaway regimes. To obtain these results, we develop a new semiclassical approach. For the Wilson-Fisher model, we propose an alternative description, which becomes weakly coupled in the large spin limit. This allows us to chart the phase diagram and obtain numerous rigorous predictions for large spin impurities in $2+1$ dimensional magnets. Finally, we also study $1/2$-BPS Wilson lines in large representations of the gauge group in rank-1 $\mathcal{N}=2$ superconformal field theories. We contrast the results with the qualitative behavior of large spin impurities in magnets.

Figures (13)

• Figure 1: An impurity of spin $s$ under $SO(3)$ interacts with the $SO(3)$ symmetric bulk. The operators $\vec{S}$ are the spin $s$ representation of $so(3)$ while the operators $\vec{\sigma}$ are the bulk spins (typically in the spin $1/2$ representation) of the nearest neighbors and the bulk Hamiltonian $H_{bulk}$ is tuned to a critical point.
• Figure 2: Phase diagram of the impurity \ref{['eq_ImpIntro']} in a free bulk theory. The blue shaded region schematically represents the one that we could reliably study with our methods. The plot is obtained from the union of the double-scaling regime, applicable for small $4-d=\varepsilon$ but arbitrary $s$, and the fixed $\varepsilon$ large $s$ region, which requires $\varepsilon s$ to be sufficiently large (see sec. \ref{['sec_free_semiclassics']} for details). The red solid line separates the region in which the theory admits two fixed points from the one in which the RG flow never terminates in a DCFT. The red dashed line is its naive extrapolation in the region that we do not control with present techniques. Notice that the extrapolation suggests that all physical impurities ($s\geq 1/2$) in $d=3$ have no stable fixed points. Finally, the purple dotted lines refer to the numerical example for $s=10$ in the main text.
• Figure 3: Regimes of applicability of different methods to capture the nontrivial DCFT fixed point of the spin impurity \ref{['eq_ImpIntro']} in the $O(3)$ WF model. We expect that this DCFT exists for all $d$ and $s$. The blue region corresponds to the standard $\varepsilon$ expansion, the red hatched region to the new semiclassical resummation method, and the green region to an effective field theory (EFT) involving the pinning field DCFT weakly coupled to a first-order $S^2$ sigma model. As we will explain in sec. \ref{['subsec_int_fix']}, the resummed perturbation approach theory is applicable for arbitrary $s$ at small $\varepsilon$, while the pinning field effective description holds for sufficiently large $\varepsilon s^2$ (including $\varepsilon\sim O(1)$, see sec. \ref{['subsec_int_anyD']} for details).
• Figure 4: Diagram that contributes the leading order term in the one-point function $\langle\phi_a^2(x)\rangle$. Dashed lines represent bulk scalar field propagators. The solid line represents the defect.
• Figure 5: Diagrams contributing the next to leading order terms in the one-point function $\langle\phi_a^2(x)\rangle$.