Dirac fermions in (2 + 1) dimensions with dynamically generated anticommuting SO(3) antiferromagnetic and ${\mathrm{Z}}_2$ Kekulé valence-bond solid (KVBS) masses map onto a field theory with a topological $\theta$ term. This term provides a mechanism for continuous phase transitions between different symmetry-broken states: topological defects of one phase carry the charge of the other and proliferate at the transition. The $\theta$ term implies that a domain wall of the ${\mathrm{Z}}_2$ KVBS order parameter harbors a spin-$1/2$ Heisenberg chain, as described by a (1 + 1)-dimensional SO(3) nonlinear sigma model with $\theta$ term at $\theta =\pi$. Using pinning fields to stabilize the domain wall, we show that our auxiliary-field quantum Monte Carlo simulations indeed support the emergence of a spin-$1/2$ chain at the ${\mathrm{Z}}_2$ topological defect. Surprisingly, the consequences of the topological term are seen far from the critical point such that the physics of apparently unrelated model systems are naturally understood by invoking them. This concept can be generalized to higher dimensions where (2 + 1)-dimensional SO(4) or SO(5) theories with topological terms are realized at a domain wall.

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• Dirac fermions
• Quantum criticality
• Quantum phase transitions
• Topological field theories
• Lattice models in condensed matter
• Quantum Monte Carlo
• Quantum field theory