  The disorder operator is often designed to reveal the conformal field theory
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)D boundary.
At bulk critical point, the effective K is suppressed but 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)D SU(2)1 CFT and (2+1)D O(3) CFT at boundary criticality. Our Letter
paves a new way to study the exotic boundary state and boundary criticality.
