Line defects in the 3d Ising model

TL;DR

The paper tests the conjecture that the twist line defect in the critical $3d$ Ising CFT flows to a conformal line defect by performing extensive lattice Monte Carlo simulations. By classifying defect-local operators under the lattice symmetry $D_8$, it extracts a low-lying spectrum of anomalous dimensions, including the exactly protected $\, abla_D$ with $\\Delta_D=2$, and the defect primaries $\\psi$ and $\\psi_{3/2}$ with $\\Delta_\\psi \approx 0.9187$ and $\\Delta_{\\psi_{3/2}} \approx 1.99$, respectively. The results also reveal scalar and higher-spin defect operators with $\\Delta_s \approx 2.27$ and $\\Delta_{t_+} \approx 3.1$, and provide universal bulk–defect ratios such as $C^\\epsilon_{\\mathbf{1}} \approx -0.167(4)$, $|C^\\sigma_\\psi| \approx 0.968(2)$, and $|C^\\sigma_{\\psi_{3/2}}| \approx 0.61(9)$. Overall, the study supports conformal invariance of the monodromy defect and opens avenues for cross-checks with bootstrap and $\epsilon$-expansion analyses.

Abstract

We investigate the properties of the twist line defect in the critical 3d Ising model using Monte Carlo simulations. In this model the twist line defect is the boundary of a surface of frustrated links or, in a dual description, the Wilson line of the Z2 gauge theory. We test the hypothesis that the twist line defect flows to a conformal line defect at criticality and evaluate numerically the low-lying spectrum of anomalous dimensions of the local operators which live on the defect as well as mixed correlation functions of local operators in the bulk and on the defect.

Figures (9)

• Figure 1: In our set-up, the domain wall $S$ is the surface across which the links are frustrated. It ends on two defect lines. We will mostly consider the proximity of one of such lines, which we take to be aligned with the $x^3$ axis.
• Figure 2: The 2d lattice of a plane transverse to the defect line. The projection of the defect plane is the heavier line, crossed by frustrated links, which plays the rôle of a $\mathbb{Z}_2$ monodromy cut.
• Figure 3: The spin variables around the monodromy line in terms of which the defect operators of lowest dimensions of table \ref{['tab:anomalous']} can be realized, as described in the text.
• Figure 4: An alternative set of spin variables in terms of which we construct some of the defect operators, as described in the text. These spins are lying around the monodromy line in plane orthogonal to it, just as in Fig. \ref{['fig:2d_4spins']}. In particular, the $\mathcal{S}$-even pseudo-scalar representation $P$ can be realized in terms of the bilinears $\sigma_i\sigma_{i+1}$ and $\tilde{\sigma}_i\tilde{\sigma}_{i+1}$, indicated in the drawing by the two sets of diagonal segments.
• Figure 5: The irrepses of the D8 dihedral group are encoded in the extended Dynkin diagram of the $\mathcal{D}_6$ algebra, as described in the text. Open circles denote one-dimensional representations while grey circles are associated to two-dimensional representations.