Physicists Capture Topology in Action at Quantum Critical Points

For decades, physicists have treated two great frameworks for describing phases of matter as if they belonged to separate worlds. Conventional phase transitions, of the kind that turns water into steam or a magnet into a demagnetized lump, are governed by order parameters that evolve continuously and by universal critical behaviour that ignores microscopic detail. Topological phase transitions, by contrast, are defined by abrupt, quantized jumps in topological invariants—integers that cannot change smoothly no matter how gently the system is deformed. The two pictures seemed fundamentally incompatible: one celebrates continuous evolution, the other demands discontinuity. A new experiment published in Nature now shows that these frameworks are not rivals but partners, and that topology leaves a measurable, characteristic imprint precisely at the critical points where phases meet.

The study, carried out by Zhi-Kang Lin, Li-Wei Wang, Ze-Lin Kong and colleagues under the guidance of Jian-Hua Jiang, Shuang Zhang and Xuejia Yu, reports experimental evidence that gapless topological states—states sitting exactly at phase boundaries—can be characterized by their entanglement spectrum and entanglement wavefunctions in both one and two dimensions. The team complemented this entanglement-based analysis with direct imaging of topological boundary modes inside a gapless bulk continuum, using carefully engineered phononic crystals as their experimental platform. The result is the first experimental observation of what the researchers call critical topology: topology that lives not inside gapped phases, but at the very transitions between them.

The conceptual key to the work is the entanglement spectrum, a tool introduced by Hui Li and F. Duncan Haldane in 2008 as a generalization of entanglement entropy. Rather than asking how much two parts of a quantum system are entangled, the entanglement spectrum asks how they are entangled, encoding the full set of eigenvalues of the reduced density matrix that describes one part alone. For gapped topological phases, theorists quickly realized that the entanglement spectrum behaves like the band structure of an edge: it carries its own boundary modes, and its topology mirrors the topology of the bulk. This Li-Haldane correspondence turned entanglement into a fingerprint of topological order, one that could identify phases that conventional symmetry-breaking analysis would miss entirely.

Extending this fingerprint to critical systems, where the energy gap closes and correlation lengths diverge, was far from straightforward. Theoretical work by Ruben Verresen, Nick Jones, Frank Pollmann and collaborators had predicted that topology and edge modes survive quantum criticality between topological insulators, and that gapless symmetry-protected topological order could exist in one-dimensional systems. Subsequent studies explored conformal boundary conditions, fidelity susceptibility at Lifshitz transitions, and universal entanglement spectra in gapless symmetry-protected states. But predictions are one thing and laboratory evidence is another. Measuring the entanglement structure of a many-body system at a critical point, where the gap vanishes and fluctuations occur at every scale, poses a formidable experimental challenge.

The Jiang and Zhang groups solved this challenge by exploiting a deep equivalence between classical wave systems and quantum systems. In a phononic crystal—an engineered structure that guides sound waves through periodic resonators—the mathematics of acoustic modes is identical to the mathematics of single-particle quantum wavefunctions. By measuring the field profiles of acoustic modes throughout the structure, the researchers could reconstruct correlation functions, and from those correlations they could compute reduced density matrices and their entanglement spectra using the standard Peschel prescription. Although the platform is classical, the entanglement quantities extracted from it are mathematically equivalent to those of the corresponding quantum system at zero temperature, and the approach is readily extensible to genuine quantum platforms such as photonic or superconducting qubit arrays.

In one dimension, the team first validated their method on gapped topological phases, showing that the entanglement spectrum correctly distinguishes phases with different topological invariants. They then tuned their structures toward the phase boundaries, where the bulk gap closes. There, the entanglement spectrum revealed a striking signature: the characteristic entanglement band structure of the critical point itself, complete with its own protected modes. The researchers also imaged the topological boundary modes directly in the gapless bulk continuum, confirming that the boundary physics predicted by the entanglement analysis appears in the real-space acoustic fields. Criticality, in other words, does not wash topology away—it preserves it in a well-defined, measurable form.

The two-dimensional experiments pushed the result further. By constructing phononic crystals whose acoustic bands mimic two-dimensional topological models, the team demonstrated that critical topology extends beyond the one-dimensional systems where theory had first anticipated it. The entanglement wavefunctions—the eigenvectors accompanying the entanglement eigenvalues—provided an even finer diagnostic, revealing the spatial and symmetry structure of the critical modes. This extension to higher dimensions matters because most technologically relevant topological phenomena, from chiral edge transport to higher-order hinge states, live in two or three dimensions, and the new results show that the entanglement toolkit travels with them.

Perhaps the most conceptually rich finding concerns what happens when different critical boundaries meet. The researchers showed that transitions among phase boundaries with distinct critical topology lead to multi-critical points in the phase diagram—special locations where several phase boundaries converge and multiple gap-closing events coincide. These multi-critical points are not accidents; they evidence a topology-driven multi-criticality and reveal a hierarchical structure in which topological phases nest inside one another like Russian dolls, with critical states forming the connective tissue between them. The phase diagram, viewed through the entanglement lens, is not a flat map of regions but a layered architecture organized by topology.

The implications reach across several fields at once. For condensed matter physicists, the work provides an experimental handle on symmetry-enriched quantum criticality and intrinsically gapless topological phases, concepts that had lived mainly in theoretical papers. For the metamaterials community, it demonstrates that classical acoustic platforms can probe questions once thought to require fully quantum experiments, extending the program the team began when they previously measured entanglement entropy and its topological signature in phononic systems. And for the broader pursuit of quantum materials, the entanglement-based characterization of critical points offers a new diagnostic that complements conventional probes such as angle-resolved photoemission or transport measurements, which struggle precisely where gaps close.

There are natural next steps. The equivalence between the classical phononic platform and the quantum limit suggests that cold atoms, trapped ions or superconducting circuits could reproduce the same measurements in settings where genuine many-body entanglement and interactions play a role. The hierarchical structure of topological phases hinted at by the multi-critical points invites a systematic classification of critical topology, much as the tenfold way once organized gapped topological phases. What the experiment establishes beyond doubt is that the boundary between the physics of phase transitions and the physics of topology was always a line drawn on paper rather than in nature. At the critical point, where a system hesitates between two orders, topology is not destroyed—it is revealed.

Subject of Research: Experimental observation of topological physics at quantum critical points using phononic crystals and entanglement spectra

Article Title: Experimental observation of critical topology

Article References: Lin, Z.-K., Wang, L.-W., Kong, Z.-L., Zhou, Y., Lin, H.-Q., Yu, X., Zhang, S., & Jiang, J.-H. (2026). Experimental observation of critical topology. Nature, 658(8135), 365-371. https://doi.org/10.1038/s41586-026-11099-x

Image Credits: AI Generated

DOI: 10.1038/s41586-026-11099-x

Keywords: critical topology, phase transitions, topological phases, entanglement spectrum, phononic crystals, quantum criticality, topological invariants, boundary modes, metamaterials, multi-criticality, gapless topological states, condensed matter physics