The disorder operator is often designed to reveal the conformal field theory (CFT) information in quantum many-body systems. By using large-scale quantum Monte Carlo simulation, we study the scaling behavior of disorder operators on the boundary in the two-dimensional Heisenberg model on the square-octagon lattice with gapless topological edge state. In the Affleck-Kennedy-Lieb-Tasaki phase, the disorder operator is shown to hold the perimeter scaling with a logarithmic term associated with the Luttinger liquid parameter $K$. This effective Luttinger liquid parameter $K$ reflects the low-energy physics and CFT for $(1+1)\mathrm{D}$ boundary. At bulk critical point, the effective $K$ is suppressed but it keeps finite value, indicating the coupling between the gapless edge state and bulk fluctuation. The logarithmic term numerically captures this coupling picture, which reveals the $(1+1)\mathrm{D}$ $\mathrm{SU}(2{)}_1$ CFT and $(2+1)\mathrm{D}$ $O(3)$ CFT at boundary criticality. Our Letter paves a new way to study the exotic boundary state and boundary criticality.

• Received 28 November 2023
• Revised 22 February 2024
• Accepted 22 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.206502