Measuring the boundary gapless state and criticality via disorder operator

The disorder operator is often designed to reveal the conformal field theory (CFT) information in the quantum many-body system. By using large-scale quantum Monte Carlo simulation, we study the scaling behavior of disorder operator 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 (AKLT) 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 keep finite value, indicating the coupling between the gapless edge state and bulk fluctuation. The logarithmic term numerically capture this coupling picture, which reveals the (1+1)d SU(2)_1 CFT and (2+1)d O(3) CFT at boundary criticality. Our work paves a new way to study the exotic boundary state and boundary criticality.