Topological tricritical Ising universality class in one dimension

Quantum critical points can host symmetry-protected topological edge modes even when the bulk is gapless, giving rise to symmetry-enriched universality classes beyond the conventional Landau--Ginzburg--Wilson paradigm. Here we show that the tricritical Ising (TCI) critical point---described by a paradigmatic conformal field theory (CFT) with emergent supersymmetry---admits a topologically nontrivial, symmetry-enriched realization, which we term the topological TCI. We construct this critical point in the cluster O'Brien--Fendley spin chain by applying a symmetry-protected topological entangler to the original O'Brien--Fendley model. The two realizations therefore share the same bulk TCI CFT. Under open boundary conditions, however, the original model exhibits the free conformal boundary condition, whereas the cluster model exhibits a spontaneously fixed boundary condition: an infinitesimal boundary field can select one of the fixed boundary states, yielding spontaneous boundary magnetization. We further show that symmetry enrichment and boundary renormalization-group flow provide two routes to the spontaneously fixed boundary condition. Whereas the free and spontaneously fixed boundary conditions of the original model can be interconverted by tuning a symmetry-preserving boundary term, no such conversion occurs for the cluster model along the boundary deformations studied here, even in the strong-deformation limit. Although they share the same conventional boundary CFT data, including the boundary operator content and boundary $g$-function, the two realizations of the spontaneously fixed boundary condition are distinguished by different time-reversal charges of the disorder field, which impose different symmetry constraints on the allowed boundary conditions under symmetry-preserving boundary deformations. For the tested symmetry-preserving boundary perturbations that lift the exact twofold degeneracy, the finite-size energy splitting is numerically consistent with exponential decay over accessible system sizes, in stark contrast to the algebraic splitting at the symmetry-enriched Ising critical point of the cluster Ising spin chain. We also find that the topological TCI realization persists along a critical line in a two-parameter phase diagram. These results establish a symmetry-enriched TCI universality class and uncover boundary distinctions invisible to conventional boundary CFT data.