  Using Quantum Monte Carlo simulations, we study the spin-1/2 Heisenberg model
on a two-dimensional lattice formed by coupling diagonal ladders. The model
hosts an antiferromagnetic N\'eel phase, a rung singlet product phase, and a
topological none trivial Haldane phase, separated by two quantum phase
transitions. We show that the two quantum critical points are all in the
three-dimensional O(3) universality class. The properties of the two gapped
phases, including the finite-size behavior of the string orders in the Haldane
phase, are studied. We show that the surface formed by the ladders ends is
gapless, while the surface exposed along the ladders is gapful, in the Haldane
phase.
  Conversely, in the gapped rung singlet phase, the former surface is gapped,
and the latter is gapless. We demonstrate that, although mechanisms of the two
gapless modes are different, nonordinary surface critical behaviors are
realized at both critical points on the gapless surfaces exposed by simply
cutting bonds without fine-tuning the surface coupling required to reach a
multicritical point in classical models. We also show that, on the gapped
surfaces, the surface critical behaviors are in the ordinary class.
