  Bulk topology and criticality can both lead to nontrivial boundary effects.
Topological orders are often characterized by their robust edge states, while
bulk critical points can have different boundary scalings governed by boundary
conditions. The interplay between these two different boundary effects is an
intriguing problem. The boundary of the deconfined quantum critical point
(DQCP) is the ideal platform for the interplay of the two boundary effects, as
the DQCP is also an intrinsically gapless symmetry protected topological
(igSPT) state. In this work we discuss the boundary of several analogues of the
DQCP. We demonstrate that the fluctuation of the bulk order parameters and
their various boundary conditions lead to a rich possibility of the edge
states, including a "pseudogap" (or super power-law decay) behavior. We also
discuss the quantum information perspective of our work, i.e. the implication
of our results on DQCP under weak-measurement. Weak-measurement followed by
post-selection can change the boundary condition at the temporal boundary in
the path-integral representation of a density matrix, which will lead to
different behaviors of the "strange correlator".
