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.

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• Quantum phase transitions
• String theory techniques in condensed matter
• Symmetry protected topological states
• Topological phase transition
• Symmetries in condensed matter
• Density matrix methods
• Finite-size scaling
• Gauge theory techniques
• Integrable systems
• Matrix product states
• Numerical Renormalization Group
• Quantum field theory (low energy)
• Tensor network methods