Impurities with a cusp: general theory and 3d Ising

TL;DR

This paper develops a general theory of cusp anomalous dimensions in conformal field theories, relating small angle behavior to defect fusion, and provides nonperturbative numerical results for the 3D Ising model, advancing understanding of impurities in quantum critical systems.

Contribution

It introduces a unified theoretical framework for cusp anomalous dimensions in CFTs and applies it to compute nonperturbative results for the 3D Ising model using fuzzy-sphere regularization.

Findings

Cusp anomalous dimension relates to defect fusion and has a concavity property.

Casimir energy between defects is always negative, indicating attraction.

Numerical results for 3D Ising model agree with theoretical predictions.

Abstract

In CFTs, the partition function of a line defect with a cusp depends logarithmically on the size of the line with an angle-dependent coefficient: the cusp anomalous dimension. In the first part of this work, we study the general properties of the cusp anomalous dimension. We relate the small cusp angle limit to the effective field theory of defect fusion, making predictions for the first couple of terms in the expansion. Using a concavity property of the cusp anomalous dimension we argue that the Casimir energy between a line defect and its orientation reversal is always negative ("opposites attract"). We use these results to determine the fusion algebra of Wilson lines in $\mathcal{N}=4$ SYM as well as pinning field defects in the Wilson-Fisher fixed points. In the second part of the paper we obtain nonperturbative numerical results for the cusp anomalous dimension of pinning field defects in the Ising model in $d=3$, using the recently developed fuzzy-sphere regularization. We also compute the pinning field cusp anomalous dimension in the $O(N)$ model at one-loop in the $\varepsilon$-expansion. Our results are in agreement with the general theory developed in the first part of the work, and we make several predictions for impurities in magnets.