  We design a (2+1)D quantum spin model in which spin-1/2 ladders are coupled
through antiferromagnetic Ising interactions. The model hosts a quantum phase
transition in the (2+1)D $Z_2$ universality class from the Haldane phase to the
antiferromagnetic Ising ordered phase. We focus on studying the surface
properties of three different surface configurations when the Ising couplings
are tuned. Different behaviors are found on different surfaces. We find
ordinary and two different extraordinary surface critical behaviors (SCBs) at
the bulk critical point. The ordinary SCBs belong to the surface universality
class of the classical 3D Ising bulk transition. One extraordinary SCBs is
induced by the topological properties of the Haldane phase. Another
extraordinary SCBs at the bulk critical point is induced by an unconventional
surface phase transition where the surface develops an Ising order before the
bulk. This surface transition is realized by coupling a (1+1)D $SU(2)_1$ CFT
boundary to a (2+1)D bulk with $Z_2$ symmetry. We find that the transition is
neither a (1+1)D $Z_2$ transition, expected based on symmetry consideration,
nor a Kosterlitz-Thouless-like transition, violating the previous theoretical
prediction. This new surface phase transition and related extraordinary SCBs
deserve further analytical and numerical exploration.
