Surface critical behavior of coupled Haldane chains

The nonordinary surface transitions at the ($2+1$)-dimensional quantum critical point precluded in corresponding classical have been found recently. mechanism of such behavior that is only dimerized Heisenberg models to date still under debate. To illuminate role symmetry-protected topological (SPT) phases inducing behaviors, we study a system on two-dimensional square lattice consisting interacting spin-1 Haldane chains, which has genuine SPT phase---the phase---at weak interchain interactions and point, belonging three-dimensional (3D) O(3) universality class, N\'eel phase. Different from previously studied models, there no dimerization current model. Cutting along chain direction or perpendicular exposes two different surfaces. Using unbiased Monte Carlo simulations, find types surfaces show completely behaviors bulk resulting states For with direction, ordinary type 3D while for nonordinary, consistent models. Our numerical results demonstrate gapless state gapped phase together mode pure scenario leads transition.