  Using Quantum Monte Carlo simulations, we study spin-1/2 diagonal ladders
coupled by ferromagnetic Heisenberg interactions. The model can also be viewed
as usual ladders with ferromagnetic rung couplings coupled by antiferromagnetic
diagonal couplings. We find that the model hosts a striped magnetic ordered
phase and two topological nontrivial Haldane phases, separated by two quantum
critical points. We show that the two quantum critical points are all in the
three-dimensional O(3) universality class irrelevant to the topological
properties of the Haldane phases. The properties of the surface formed by
ladder ends in the two Haldane phases are studied. We find that the surface
states are both gapless due to the symmetry-protected topological bulk states.
We further demonstrate that extraordinary surface critical behaviors are
realized at both critical points on such gapless surfaces without enhancing the
surface coupling. Notably, the surface is not expected to be ordered in the
three-dimensional classical O(3) critical point, suggesting that the
topological properties of the Haldane phases are responsible for such surface
critical behavior.
