  The boundary critical behavior of the two-dimensional (2D) quantum
antiferromagnetic(AF) XXZ model coupled with either a dangling spin-1/2 XXZ
chain or a dangling two-leg ladder on the boundary is studied with the
bosonization and renormalization group analysis. A rich boundary phase diagram
is obtained in each case. In the dangling chain case, the boundary either
develops a long-range AF order in the easy-plane or the easy-axis direction
with extraordinary critical behavior, or has a valence bond solid order with
ordinary boundary critical behavior. In the case of a dangling two-leg ladder,
besides the possible easy-plane or easy-axis AF ordered phases on the boundary
with extraordinary critical behavior, the boundary can form a singlet phase
with ordinary critical behavior without breaking any symmetry.
