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The disorder operator is often designed to reveal the conformal field theory(CFT) information in the quantum many-body system. By using large-scale quantumMonte Carlo simulation, we study the scaling behavior of disorder operator onthe boundary in the two-dimensional Heisenberg model on the square-octagonlattice with gapless topological edge state. In the Affleck-Kennedy-Lieb-Tasaki(AKLT) phase, the disorder operator is shown to hold the perimeter scaling witha logarithmic term associated with the Luttinger Liquid parameter K. Thiseffective Luttinger Liquid parameter K reflects the low energy physics and CFTfor (1+1)d boundary. At bulk critical point, the effective K is suppressed butkeep finite value, indicating the coupling between the gapless edge state andbulk fluctuation. The logarithmic term numerically capture this couplingpicture, which reveals the (1+1)d SU(2)_1 CFT and (2+1)d O(3) CFT at boundarycriticality. Our work paves a new way to study the exotic boundary state andboundary criticality.

Measuring the boundary gapless state and criticality via disorder operator