To appreciate why this breakthrough matters, it helps to revisit what topology means in modern physics. Unlike conventional phases of matter, whose identities are dictated by symmetry, topological phases are characterized by global properties that remain unchanged under continuous deformation. The classic analogy compares a coffee mug and a doughnut: each possesses exactly one hole, and one can be smoothly reshaped into the other without tearing or gluing. Because these global properties are quantized, they resist small perturbations. The quantum Hall effect and topological insulators, whose discovery was recognized by the 2016 Nobel Prize in Physics, exploit this robustness to sustain conducting edge states that persist even in the presence of impurities and disorder, so long as the underlying symmetry is preserved.

That robustness, however, has always come with a strict prerequisite. The topological invariant that protects edge states is only well defined when the bulk of the material possesses an energy gap. Close the gap, the standard reasoning goes, and the invariant loses meaning, the edge states disappear, and the phase loses its topological character. This assumption has been so deeply embedded in the field that the very definition of a topological phase has been tied to gapped systems. Generations of researchers have treated the energy gap not as a convenience but as an essential ingredient of topological order.

Criticality occupies the opposite corner of the physics landscape. At a critical point, such as the boiling of water into vapor or the loss of magnetism at the Curie temperature, the energy gap closes completely, microscopic details are washed out, and only long-wavelength collective behavior remains. Remarkably, materials with entirely different microscopic structures can exhibit identical critical exponents at such points, placing them in the same universality class. This universality, honored by the 1982 Nobel Prize in Physics, is one of the most profound insights in science: near criticality, diverse systems converge on a single universal description. But that description is continuous and fluctuation-dominated, while topology is discrete and invariant-protected. For decades the two frameworks were regarded as separate domains that developed independently with little overlap.

Challenging that boundary required years of theoretical groundwork. The question of whether topology could survive at a critical point was once considered a fundamental challenge to the conventional understanding of topological phases. In recent years, Prof. Yu and collaborators systematically explored the possibility through a series of theoretical studies. In 2022 they proposed a classification theory for topology at quantum critical points, published in Physical Review Letters. In 2024 they introduced a generalized topological bulk–edge correspondence applicable to critical points, and in 2026 they extended the theory to nonequilibrium dynamics, work highlighted as an Editor’s Suggestion. Together with Prof. Limei Xu of Peking University and Chair Professor Hai-Qing Lin of Zhejiang University, Yu also authored the first comprehensive review of the field in Physics Reports, consolidating the complete theoretical framework of critical topology.

Turning that framework into laboratory reality posed an entirely different order of difficulty. To realize critical topology in a physical system, researchers must tune the system with extreme precision to a critical point where the bulk becomes completely gapless, and then detect the signatures of topological edge states against strong fluctuations and background noise. The challenge has been compared to stopping a high-speed train exactly at the edge of a cliff while still measuring a tiny speck of dust on one of its wheels. For years, the predictions of critical topology therefore remained confined to theoretical studies and numerical simulations, with no experimental platform able to meet the twin demands of critical tuning and topological resolution.

That changed with two experiments built on acoustic metamaterials, artificial structures in which sound waves mimic the behavior of quantum systems. In the first study, titled Observation of critical topological phase transition, the team led by Prof. Baile Zhang at Nanyang Technological University observed topological zero-energy modes and π modes at a one-dimensional Floquet critical point. The experiment relied on a periodically driven dimerized lattice, the Floquet SSH α-chain, in which two distinct half periods alternately host only next-nearest-neighbor and nearest-neighbor couplings. In the acoustic implementation, a waveguide array maps the propagation direction of sound onto an effective time coordinate, with straight connecting tubes engineering nearest-neighbor coupling and curved tubes implementing next-nearest-neighbor coupling. The work grew out of a theoretical framework proposed by Yu and developed further with collaborators, published in Communications Physics in 2025. Co-first authors are Dr. Zheyu Cheng of Nanyang Technological University and Dr. Xiuhai Zhang of Northwestern Polytechnical University, with co-corresponding authors including Yu, Prof. Longwen Zhou of Ocean University of China, Prof. Jiangbin Gong of the National University of Singapore, and Prof. Zhang.

The second study, titled Experimental observation of critical topology, took a fundamentally different route. The team led by Prof. Jianhua Jiang at the University of Science and Technology of China applied the generalized Li–Haldane bulk–edge correspondence proposed by Yu in recent publications in Physical Review B and Physical Review Research. Rather than probing the energy spectrum directly, the experimenters measured the entanglement spectrum, a quantity drawn from quantum information theory that encodes the pattern of quantum correlations in a system. Using a pump-probe protocol on a one-dimensional phononic crystal, they measured the local response, extracted band dispersions and Bloch wavefunctions through singular-value decomposition, constructed a k-space correlation matrix, and transformed it into real space to obtain the entanglement spectrum. In the nontrivial case, a pair of mid-gap modes at exactly half filling appeared, providing direct experimental evidence for nontrivial critical topology in both one and two dimensions. Co-first authors are Dr. Zhikang Lin of The University of Hong Kong, Mr. Liwei Wang of the University of Science and Technology of China, and Dr. Zelín Kong of Soochow University, with co-corresponding authors Yu, Prof. Shuang Zhang of The University of Hong Kong, and Prof. Jiang.

The significance of having two papers in the same journal issue, built on two independent experimental platforms with different design strategies and measurement approaches, yet resting on the same theoretical origin, is difficult to overstate. Together they provide the first experimental evidence in real physical systems that topology can survive even when the energy gap closes, and that critical universal behavior and topological properties can coexist in the same system. Prof. Yu, serving as co-corresponding author on both papers, provided key theoretical guidance spanning modeling, mechanism interpretation, and the physical picture underlying both experiments. The convergence of independent experimental confirmation with a unified theoretical framework transforms critical topology from a speculative mathematical possibility into an established empirical phenomenon.

The implications extend well beyond the acoustic platforms used in these demonstrations. Critical topology opens new avenues for topological quantum computation, where robust states at criticality could offer novel protection schemes, for the design of new acoustic devices that exploit topology without requiring gapped bulk bands, and for the study of nonequilibrium topological phases of matter, where driven and fluctuating systems are the norm rather than the exception. Topology and criticality, two pillars of physics long considered fundamentally incompatible, have now met in the laboratory, and the theoretical force behind that convergence came from the Eastern Institute of Technology, Ningbo. The research group led by Prof. Yu, which focuses on quantum phase transitions and critical phenomena in strongly correlated many-body systems using tools ranging from conformal field theory to tensor networks and quantum Monte Carlo simulations, has spent the past five years at the forefront of this emerging field, and these two Nature papers represent its most striking vindication to date.

Subject of Research: Experimental observation of critical topology in acoustic metamaterials

Article Title: Breaking a long-standing boundary in physics: Xue-Jia Yu’s team and collaborators publish two consecutive papers in Nature, revealing that topology can survive at critical points

Article References: Breaking a long-standing boundary in physics: Xue-Jia Yu’s team and collaborators publish two consecutive papers in Nature, revealing that topology can survive at critical points. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: critical topology, topological phases, phase transitions, acoustic metamaterials, Floquet systems, entanglement spectrum, bulk-edge correspondence, energy gap, quantum criticality, Nature, Xue-Jia Yu, condensed matter physics

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

The appeal of quantum batteries lies in exploiting resources that classical devices simply do not have. Entanglement, quantum coherence, and many-body correlations can be harnessed to accelerate charging beyond classical transport limits. Since Alicki and Fannes formalized the concept more than a decade ago, theorists have demonstrated that collective charging protocols can push charging power to scale superextensively with system size—approaching the fundamental Heisenberg bound, where power grows quadratically with the number of cells rather than linearly. Experiments using spin systems, superconducting circuits, and organic molecular aggregates have begun validating these predictions, turning quantum batteries into a genuine technological frontier for powering quantum computers, sensors, and communication networks at the nanoscale.

The Iranian team focused on spin-1/2 chains because they are among the most tunable and theoretically tractable platforms for studying quantum energy storage. Their model is an anisotropic XY chain augmented with a three-spin interaction whose strength and anisotropy can be varied independently. The three-spin term is more than a decorative addition: it breaks the continuous rotational symmetry of the ordinary XX chain and enriches the ground-state phase diagram dramatically. In the isotropic limit, the model supports six distinct non-trivial topological phases, each labeled by a winding number of plus or minus one or two. Switching on XY anisotropy adds a seventh, topologically trivial region and a fifth critical line separating phases whose bulk excitation spectra and edge physics differ fundamentally.

A key technical achievement of the study is that the model remains exactly solvable. Using the Jordan–Wigner transformation, the authors map the interacting spin chain onto a quadratic form of non-interacting fermions—a generalized version of the celebrated Kitaev chain, the paradigmatic model of one-dimensional topological superconductivity. But where the standard Kitaev model has only nearest-neighbor hopping and p-wave pairing, this generalized chain includes next-nearest-neighbor hopping and long-range p-wave pairing, producing a far richer landscape of topological phases. In momentum space, the Hamiltonian decouples into independent two-level blocks for each momentum mode, allowing the researchers to diagonalize it exactly and track every quasiparticle excitation through the charging process.

The charging protocol itself is a quantum quench. The battery is prepared in the ground state of an initial Hamiltonian, which serves as its reference state. At time zero, the Hamiltonian is abruptly switched to a different charging Hamiltonian, and the system evolves unitarily, absorbing energy as quasiparticle modes are populated. After a charging duration, the Hamiltonian is quenched back, decoupling the battery from its charger and trapping the stored energy. This sudden-switch protocol is one of the most natural ways to drive a quantum many-body system out of equilibrium, and it directly probes how the spectral structure of the underlying chain governs energy absorption.

From the exact solution, the authors derive an analytical expression for the stored energy as a sum over momentum modes, each contributing an oscillatory term weighted by the overlap between the initial and final quasiparticle states. This structure defines a geometric factor—an upper bound on the maximum energy the battery can ever hold for a given quench. The team’s central discovery is that when the system evolves for a long time, the dephasing between the different momentum modes drives each oscillatory contribution toward its time average of one half. The steady-state stored energy therefore equals exactly half of the geometric factor, universally, across every quench protocol and every parameter regime they examined. This half-energy rule reflects a rigid geometric constraint imposed by the Hilbert-space structure of the fermionized chain, and numerical simulations on chains of one thousand spins confirm the analytical prediction with striking precision.

Perhaps the most striking result is how sensitive this energy storage capacity is to quantum criticality. The geometric factor exhibits a pronounced maximum precisely at one of the model’s critical points, where the bulk energy gap closes and the system undergoes a topological phase transition. The first derivative of the geometric factor develops sharp cusps at every critical value, revealing that charging efficiency is deeply rooted in the critical fluctuations of the ground state. Moreover, whichever critical line maximizes the stored energy depends on the initial configuration of the quench, meaning the battery’s performance encodes detailed information about where its charging trajectory begins and ends within the topological phase diagram. In effect, the charged battery acts as a readout of the quantum phase transitions it was driven across.

The short-time charging dynamics are equally revealing. A Taylor expansion shows that the stored energy initially grows quadratically with time, with a rate set by the initial quasiparticle dispersion, the square of the final dispersion, and the transition probabilities between eigenstates. The time to reach peak energy is governed by a ratio of spectral moments of the initial and final Hamiltonians, and because the quench populates a broad swath of the excitation spectrum, this peak time varies smoothly and robustly across the parameter range rather than spiking at isolated resonances. The team supplemented their analytical estimate with an empirical polynomial fit that accurately captures the numerical peak time over the full quench interval, providing a practical benchmark for the charging performance of this architecture.

The interplay between topology and coherence becomes even clearer at long times. In generic quenches, energy disperses across many modes, decoherence accumulates, and the post-revival energy maximum falls short of the initial charging peak. But when the final Hamiltonian sits at a critical point, the gap closing enhances mode degeneracy and slows the decay of coherence, allowing quantum revivals to reconstruct stored energy more effectively. At some—but not all—critical points, the revival maximum actually exceeds the short-time peak, a signature of topological protection mitigating decoherence. Meanwhile, the peak charging power density rises essentially monotonically as the quench drives the system deeper into different topological regimes, showing that the location of the final Hamiltonian in the phase diagram directly controls how fast the battery can be charged.

The authors emphasize that their model could plausibly be engineered on programmable quantum simulation platforms, including Rydberg-atom arrays, where recent experiments have demonstrated tunable spin Hamiltonians with controllable anisotropies and Floquet-engineering schemes have been proposed for realizing generalized spin-exchange interactions. A full experimental implementation of the three-spin Hamiltonian remains beyond the scope of the current work, but the framework establishes a concrete route toward quantum batteries whose performance is deliberately steered by topological design. Beyond energy storage, the exquisite sensitivity of charging dynamics to critical lines suggests a dual use: such devices could simultaneously function as precision probes of quantum phase transitions, turning a future quantum battery into both a power source and a diagnostic instrument for the quantum materials it is built from.

Subject of Research: Energy storage limits and charging dynamics in an anisotropic three-spin interaction quantum battery

Article Title: Energy storage limits and criticality in an anisotropic three-spin interaction quantum battery

Article References: Beygi, V. Y., Mahdavifar, S., Mirmasoudi, F., & Ahadpour, S. (2026). Energy storage limits and criticality in an anisotropic three-spin interaction quantum battery. Results in Physics, 88, Article 108748. https://doi.org/10.1016/j.rinp.2026.108748

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108748

Keywords: quantum battery, three-spin interaction, topological phase transition, spin chain, quantum criticality, ergotropy, quantum quench, charging power, Jordan-Wigner transformation, Kitaev chain, stored energy, Rydberg atoms