Unconventional Surface Critical Behaviors Induced by Quantum Phase Transition from Two-Dimensional Affleck-Kennedy-Lieb-Tasaki Phase to N\'eel Order

A symmetry-protected topological phase has nontrivial surface states in the presence of certain symmetries, which can either be gapless or be degenerate. In this work, we study the physical consequence of such gapless surface states at the bulk quantum phase transition (QPT) that spontaneously breaks these symmetries. The two-dimensional Affleck-Kennedy-Lieb-Tasaki phase on a square lattice and its QPTs to N\'eel ordered phases are realized with the spin-$1/2$ Heisenberg model on a decorated square lattice. With large-scale quantum Monte Carlo simulations, we show that even though the bulk QPTs are governed by the conventional Landau phase transition theory, the gapless surface state induces unconventional universality classes of the surface critical behaviors.