Non-Factorizing Interface in the Two-Dimensional Long-Range Ising Model

Abstract

The factorization proposal claims that the co-dimension one "pinning defect", on which a local relevant operator is integrated, factorizes the space into two halves in general conformal field theories in the infrared limit. In this letter, we study a two-dimensional long-range Ising model at criticality with a line defect or an interface, which physically corresponds to changing the local temperature on it. We show that in the perturbative regime, it is not factorizing even in the infrared limit. An intuitive explanation of the non-factorization is that the long-range Ising model is equivalent to a local conformal field theory in higher dimensions. In this picture, the space is still connected through the "extra dimension" across the defect line.

Figures (3)

• Figure 1: Diagrams contributing to the one-loop order of the one-point function $\langle \Phi^4(x) \rangle$, with the solid square being the vertex of the quartic coupling $h_0$.
• Figure 2: Loop-like diagrams contributing to the one-point function $\langle \Phi^2(x) \rangle$, with the solid dot representing the vertex of the quadratic coupling $g_0$.
• Figure 3: The mixed diagram contributing to $\langle \Phi^2(x) \rangle$ at the second order in the couplings, with the square and dot representing the vertices of $h_0$ and $g_0$ respectively.