  We study the interplay between two nontrivial boundary effects: (1) the two
dimensional ($2d$) edge states of three dimensional ($3d$) strongly interacting
bosonic symmetry protected topological states, and (2) the boundary
fluctuations of $3d$ bulk disorder-to-order phase transitions. We then
generalize our study to $2d$ gapless states localized at an interface embedded
in a $3d$ bulk, when the bulk undergoes a quantum phase transition. Our study
is based on generic long wavelength descriptions of these systems and
controlled analytic calculations. Our results are summarized as follows: ($i.$)
The edge state of a prototype bosonic symmetry protected states can be driven
to a new fixed point by coupling to the boundary fluctuations of a bulk quantum
phase transition; ($ii.$) the states localized at a $2d$ interface of a $3d$
SU(N) quantum antiferromagnet may be driven to a new fixed point by coupling to
the bulk quantum critical modes. Properties of the new fixed points identified
are also studied.
