Observation of critical topological phase transition.

Phase transitions are traditionally understood within two seemingly incompatible theoretical paradigms. Landau-Ginzburg phase transitions1,2 are continuous, occurring at critical points where microscopic details become irrelevant and universal behaviour emerges, driven by gapless fluctuations with scale-invariant correlations3. By contrast, topological phase transitions connect gapped phases characterized by quantized topological invariants4,5 and, in doing so, necessarily pass through gapless phase boundaries where such invariants become ill-defined6-9. Here using an acoustic metamaterial platform, we experimentally demonstrate a unifying scenario-a critical topological phase transition-in which topology becomes well-defined at criticality10-13, overcoming this conventional incompatibility, and reorganizes critical behaviour by driving a phase transition along critical phase boundaries14. This transition gives rise to a topology-enforced multicritical point at an isolated intersection of critical boundaries with distinct topology. Along these boundaries, we observe topological edge states and diagnose criticality by means of logarithmic entanglement-entropy scaling15-17. At the multicritical point, the entanglement entropy shows additive logarithmic scaling given by the sum of contributions from adjacent boundaries, directly evidencing topology-enforced multicriticality. These findings establish a hierarchical organization of phase transitions, in which topology shapes criticality beyond the conventional Landau-Ginzburg paradigm and standard topological phase transitions.