  Decades of research have revealed a deep understanding of topological quantum
matter with protected edge modes. We report that even richer physics emerges
when tuning between two topological phases of matter whose respective edge
modes are incompatible. The frustration at the edge leads to novel boundary
physics, such as symmetry-breaking phases with exotic non-Landau transitions --
even when the edge is zero-dimensional. As a minimal case study we consider
spin chains with $\mathbb{Z}_3 \times \mathbb{Z}_3$ symmetry, exhibiting two
nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D
critical transition between these SPT phases, we find two stable 0+1D boundary
phases, each spontaneously breaking one of the $\mathbb{Z}_3$ symmetries.
Furthermore, we find that a single boundary parameter tunes a non-Landau
boundary critical transition between these two phases. This constitutes a 0+1D
version of an exotic phenomenon driven by charged vortex condensation known as
deconfined quantum criticality. This work highlights the rich unexplored
physics of criticality between nontrivial topological phases and provides
insights into the burgeoning field of gapless topological phases.
