Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line

TL;DR

This work uses the folding trick to map a 2D critical Ising model with a defect line to a boundary problem in a c=1 theory, enabling a complete boundary-state classification via the Z2 orbifold of the free boson (Ashkin–Teller at the decoupling point). It identifies two continuous defect families (Dirichlet and a novel von Neumann class) and eight discrete defect classes by matching to orbifold boundary states, and derives the full spectrum of boundary operators, exact boundary two-spin correlators, and a universal term in the defect free energy. The results reproduce known Ising-defect behavior in limiting cases and provide a unified CFT framework for classifying defect universality classes, RG flows (via the g-theorem), and dualities involving order/disorder operators. The approach also clarifies connections to generalized defects studied previously and predicts a complete set of universal defect behaviors in the Ising model, with potential extensions to general Ashkin–Teller and orbifold CFTs.

Abstract

We study the critical two-dimensional Ising model with a defect line (altered bond strength along a line) in the continuum limit. By folding the system at the defect line, the problem is mapped to a special case of the critical Ashkin-Teller model, the continuum limit of which is the $Z_2$ orbifold of the free boson, with a boundary. Possible boundary states on the $Z_2$ orbifold theory are explored, and a special case is applied to the Ising defect problem. We find the complete spectrum of boundary operators, exact two-point correlation functions and the universal term in the free energy of the defect line for arbitrary strength of the defect. We also find a new universality class of defect lines. It is conjectured that we have found all the possible universality classes of defect lines in the Ising model. Relative stabilities among the defect universality classes are discussed.

Figures (5)

• Figure 1: The Ising model the defect line on a cylinder. The defect line is parallel to the cylinder axis.
• Figure 2: The folding of the Ising model on a cylinder to a $c=1$ theory on a strip. We fold at the defect line and also at the line on the opposite side. These lines correspond to the boundary in the folded system.
• Figure 3: The location of spin operators used in the graph of correlation functions (Fig. \ref{['fig:graph']}). We fix the distance of spin operators from the boundary to $1$, and leave the horizontal distance $r$ between them as a variable.
• Figure 4: The two-spin correlation for various strength of the defect. We show the result on (a) $\langle \sigma_1 \sigma_1 \rangle$ and (b) $\langle \sigma_1 \sigma_2 \rangle$ for $\varphi_0 = 0$ (strong coupling and anisotropic limit), $0.1 \pi , 0.2 \pi , 0.25 \pi$ (no defect) , $0.3 \pi , 0.4 \pi$ and $0.5 \pi$ (free boundary condition). They are shown as a function of the distance $r$, in a log-log plot.
• Figure 5: The correlation function of two spins located symmetrically about the defect line. It is characterized by the distance from the line $y$ and the angle $\theta$.