The reactions in question are among the most celebrated in all of chemistry. In 1965, R. B. Woodward and Roald Hoffmann formulated their rules for the conservation of orbital symmetry, explaining why some pericyclic reactions proceed effortlessly while others, apparently similar, are essentially impossible under thermal conditions. The rules classify reaction pathways as symmetry allowed or symmetry forbidden depending on whether the energies of the relevant molecular orbitals cross as the molecule contorts along its reaction coordinate. A crossing of orbitals of different symmetry signals a forbidden pathway; an avoided crossing between orbitals of matching symmetry signals an allowed one. These selection rules earned Hoffmann a share of the Nobel Prize in Chemistry and have steered synthetic organic chemistry ever since.

Yet the Woodward-Hoffmann framework rests on a quiet assumption inherited from molecular orbital theory: that each electron can, to a first approximation, be treated as moving in the average field of all the others. Chemists have long known that this single-particle picture fails near certain geometries. As a bond forms or breaks, electrons can become strongly correlated, their motions entangled in ways that no independent-particle description can capture. In such regimes the very concept of a molecular orbital energy band, and with it the topological invariants that classify band structures, loses its footing. The topological band theory that works so beautifully for crystals assumes well-defined quasiparticles and a noninteracting or weakly interacting reference system, and neither survives the strong static correlation that plagues, for example, the automerization of cyclobutadiene or the ring opening of butadiene.

The Penn State team’s solution is to abandon orbitals as the fundamental objects and work instead with the single-particle Green’s function, a quantity that encodes all the information about how an electron propagates through the interacting many-electron system. In the language of Green’s functions, the familiar features of band theory reappear as poles: sharp peaks where the function blows up, corresponding to well-defined quasiparticle excitations. But Green’s functions possess a second, less familiar kind of feature: zeros, points where the function vanishes entirely. In strongly correlated systems, zeros are not mathematical curiosities. They can carry topology in their own right, a fact that condensed-matter physicists have exploited in recent years to understand Mott insulators and other systems where interactions dominate. The new work shows that this duality is exactly what chemistry needs.

The central result is strikingly clean. When the researchers computed the Green’s functions of reacting molecules along symmetry-forbidden pathways, they found that the obstruction to the reaction does not appear as a crossing of poles, as molecular orbital theory would predict, but as a crossing of zeros. In other words, the forbidden nature of the pathway is encoded in the vanishing of the electron propagation amplitude rather than in the divergence of an orbital energy. For symmetry-allowed pathways, by contrast, the poles behave in the familiar way, avoiding each other as the orbitals of matching symmetry mix. The distinction between poles and zeros thus becomes a many-body diagnostic that separates dynamic correlation, which perturbs quasiparticle energies, from static correlation, which destroys them and replaces pole crossings with zero crossings.

To make this diagnostic quantitative, the authors constructed symmetry-resolved topological invariants: integer quantities computed from the Green’s function within each irreducible representation of the molecular symmetry group. These invariants play the role that Chern numbers and symmetry indicators play in topological insulators, but they are defined for finite molecules along a reaction coordinate rather than for crystals in momentum space. Crucially, they are end-point diagnostics: one computes the invariant at the reactant geometry and at the product geometry, and if the values differ, the pathway is topologically forbidden. No continuous, symmetry-preserving deformation of the electronic structure can connect the two ends, because the invariant can only change when the Green’s function develops a singularity, which is precisely the zero crossing that characterizes the forbidden route.

The framework was put to the test on textbook systems. In benzene and cyclobutadiene, the pole-versus-zero analysis cleanly distinguished dynamic from static correlation, with cyclobutadiene’s notorious diradical character manifesting as Green’s function zeros. Along the electrocyclic ring closure of butadiene to cyclobutene, the conrotatory and disrotatory pathways, one allowed and one forbidden under the Woodward-Hoffmann rules, displayed exactly the expected behavior: poles avoided crossing on the allowed route, while zeros crossed on the forbidden one. The team also examined a 4π photoswitch, a molecular system relevant to the design of light-driven molecular machines, and found that the zero crossings persist even when the molecular symmetry is weakly broken, suggesting that the topological signature is robust enough to matter in realistic, slightly asymmetric chemical environments.

The work forges a bridge between two research communities that have largely talked past each other. On one side stands the topological quantum chemistry program, which since 2017 has catalogued how band representations and symmetry indicators classify crystalline materials, and its earlier molecular antecedents, including topological Chern indices in molecular spectra and Muechler’s own work on perplectic structures in molecules. On the other side stands the physics of strong interactions, where Green’s function zeros have been shown to underlie the breakdown of Luttinger’s theorem, to generate emergent quasiparticles at Luttinger surfaces, and to produce topological phase transitions in Mott insulators without any gap closing of single-particle states. The molecular setting turns out to be an ideal laboratory in which these ideas meet, because reaction coordinates provide a natural one-dimensional parameter, and molecular symmetry groups supply the classification machinery.

There are practical implications as well. Selection rules are the currency of chemical intuition, and a many-body generalization of them promises to extend that intuition to reactions where single-reference methods fail. The authors suggest that tracking Green’s function zeros directly could become a computational tool for predicting whether a proposed pathway is viable, particularly for photochemistry and for the design of molecular switches, where static correlation is endemic. The distinction between dynamic and static correlation, currently a source of endless trouble for quantum chemistry methods, gains a sharp spectral definition: dynamic correlation moves poles, static correlation creates zeros. That kind of crisp criterion could guide the choice of electronic structure methods before expensive calculations are attempted.

The study also resonates with a broader theme in contemporary physical chemistry: the growing recognition that geometric and topological phases shape molecular dynamics. Berry phases around conical intersections, geometric-phase effects in nonadiabatic dynamics, and even direct experimental observations of geometric-phase interference in molecules have all matured over the past decade. The new formalism adds a complementary piece, showing that topology does not merely decorate molecular motion but can fundamentally determine which reactions are possible. What began as a set of empirical rules drawn on blackboards in the 1960s now stands revealed as a special case of a deep many-body principle, one in which the fate of a chemical reaction is written not in the energies of orbitals but in the zeros of a Green’s function. For a field that has spent sixty years reasoning with orbital pictures, that is a genuinely new way of seeing an old problem.

Subject of Research: Topological classification of orbital-symmetry-controlled chemical reactions using many-body Green's functions

Article Title: Topological transitions in orbital-symmetry-controlled chemical reactions

Article References: Xie, Z., Mirzanejad, A., & Muechler, L. (2026). Topological transitions in orbital-symmetry-controlled chemical reactions. Nature Physics. https://doi.org/10.1038/s41567-026-03455-5

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03455-5

Keywords: topological band theory, Green's functions, Woodward-Hoffmann rules, orbital symmetry, strong correlation, chemical reactions, molecular orbital theory, topological invariants, pericyclic reactions, Mott insulators, quantum chemistry, Nature Physics

The researchers find quantum anomalous Hall insulators with absolute Chern numbers spanning |C| = 1 up to |C| = 7 near a moiré filling factor v = 1 and again around v ≈ 3. In topological band language, the Chern number counts how many chiral edge channels the system supports, and higher-|C| phases imply more intricate internal topology. Achieving such high-Chern insulating states in a tunable lattice system strengthens the case that moiré engineering can emulate—and extend—the physics usually associated with Landau levels.

Most compelling is the emergence of a fractional Chern insulator with C = 7/3 near v = 2/3. Fractional Chern insulators are the lattice analog of fractional quantum Hall phases: they host fractionally charged quasiparticles and can support anyonic exchange statistics. While many theoretical and experimental studies have focused on fractional states tied to known “fractional quantum Hall-like” sequences, this C = 7/3 state lies beyond commonly discussed patterns derived from the Jain sequence or high-Chern constructions. The result therefore points to a richer hierarchy of fractional topology in multi-Chern flat bands than previously catalogued.

The broader implication is that the system provides a route to probing fractionally charged excitations without relying on a strong magnetic field. In conventional fractional quantum Hall physics, the Landau level framework constrains both the allowed fractions and the structure of excitations. Here, the moiré flat-band setting replaces that basis, suggesting that lattice geometry and band topology can generate new excitation categories—potentially including anyons with properties distinct from their Landau-level counterparts.

By demonstrating a high-|C| fractional phase at a specific moiré filling, the study expands the experimental map of topological flat-band matter. It also motivates future measurements aimed at extracting quasiparticle charge, characterizing edge-mode structure, and testing how fractional statistics manifest in high-Chern fractional states. If such states can be reliably stabilized and controlled, moiré graphene may become an increasingly powerful platform for anyon research and topological quantum design.

Subject of Research: Fractional high-Chern insulators in twisted rhombohedral graphene moiré systems

Article Title: Fractional high-Chern insulator in twisted rhombohedral graphene.

Article References: Li, Z., Wang, W., Wang, F. et al. Fractional high-Chern insulator in twisted rhombohedral graphene. Nature (2026). https://doi.org/10.1038/s41586-026-10762-7

Image Credits: AI Generated

DOI: https://doi.org/10.1038/s41586-026-10762-7

Keywords: fractional Chern insulator; high-Chern number; quantum anomalous Hall; moiré flat bands; twisted rhombohedral graphene; anyonic excitations

At the core of this investigation lies a two-dimensional time-modulated lattice subject to nonlinear effects where solitons serve as the primary agents of transport. The researchers demonstrate that the soliton’s net displacement over a complete driving cycle is not arbitrary but is topologically quantized. This quantization stems from distinct Chern numbers, which are fundamental topological invariants traditionally associated with band theory in condensed matter physics. Crucially, the work extends beyond the established first Chern number—typical of one-dimensional linear pumps—introducing higher-order Chern invariants that govern transport in multi-dimensional, nonlinear systems.

Topological pumping refers to the phenomenon where a wave packet or particle systematically shifts across a lattice as a system parameter evolves cyclically in time. In linear regimes, this transport is discretized and quantified by an integer number corresponding to a first Chern number, reflecting the global topological properties of the band structure. However, introducing nonlinearity into such driven lattices significantly enriches the transport dynamics. Here, the soliton pumping is influenced by multiple Chern numbers in higher dimensions, including second Chern numbers, which offer a refined classification of the soliton’s quantum transport behavior in the two-dimensional lattice.

The nonlinear dynamics carve out distinct transport regimes. In one regime, solitons exhibit integer-quantized motion, moving an exact integer multiple of unit cells per driving cycle. This integer displacement is intricately tied to the quantization dictated by the cumulative Chern invariants of the system’s underlying topological bands. In an alternative regime, the researchers uncover fractional-quantized soliton pumping, where the displacement per cycle appears as a rational fraction of the lattice constant. This fractional quantization signals the emergence of subtle topological phases and nonlinear effects coalescing to produce transport phenomena not explained by conventional linear theories.

Beyond quantization, the soliton’s mobility is sensitive to the lattice band structure and the strength of nonlinear interactions. At strong nonlinearities, solitons become localized, entering a trapped regime wherein their position remains nearly stationary throughout the driving period. This nontrivial localization hints at a competition between nonlinear self-focusing effects and topological driving forces. Furthermore, anisotropic transport behavior was observed, where soliton displacement differs along perpendicular spatial directions. Such anisotropy results in complex mixed regimes featuring different topological quantization on different lattice axes, adding layers of control in engineering wave-packet motion through nonlinear lattices.

To experimentally confirm these theoretical predictions, the team designed nonlinear topolectrical circuits mimicking the time-modulated lattice dynamics with inherent nonlinearity. These topolectrical circuits, composed of nonlinear circuit elements arranged in time-varying networks, serve as versatile platforms to emulate the nonlinear wave dynamics and measure soliton transport properties with high fidelity. The experiments successfully captured integer and fractional quantized soliton pumping, the onset of soliton trapping, and anisotropic transport phenomena, affirming the theoretical framework and the robustness of topological invariants in nonlinear settings.

The implications of this work stretch far beyond the immediate physical system studied. By revealing how higher-order topological invariants dictate nonlinear wave transport, it opens new avenues to control and harness localized excitations in various engineered media. Topological concepts traditionally confined to linear, electronic systems now find application in nonlinear optics, acoustics, and circuit platforms, where dynamic control over wave localization and pumping can lead to breakthroughs in signal processing, energy delivery, and quantum information transfer.

Delving into the mathematical structure, the involvement of higher-dimensional Chern numbers corresponds to sophisticated geometric phases accumulated by the soliton’s wavefunction during one driving cycle in parameter space. These phases encode global topological information inaccessible through local band parameters alone. The addition of nonlinearity effectively couples the soliton’s internal degrees of freedom to the geometry of the lattice’s topological bands, resulting in a rich tapestry of dynamical responses modulated by these topological invariants.

Furthermore, the fractional quantization regime represents a subtle form of topological pumping where the soliton’s trajectory embodies a rational winding number. This regime challenges conventional understandings based predominantly on linear theory and integer-valued invariants, suggesting that nonlinearities and multi-dimensional topology may host unexplored fractionalized transport phenomena. Understanding these effects could illuminate parallels with fractional quantum Hall states and other exotic topological phases in condensed matter physics.

The study also emphasizes the precision with which topological invariants control not only the magnitude but the directionality of the soliton’s movement. The observed anisotropic pumping behavior hints at the possibility of designing waveguiding devices where solitons can be steered along preferred lattice directions by tuning lattice parameters or nonlinear interactions. Such controllability adds functional versatility to topological insulator analogs in nonlinear regimes, enabling purpose-built pathways for information or energy transmission.

In summary, this research pioneers a new paradigm where nonlinear wave physics, high-dimensional topology, and artificial lattice engineering converge to produce controlled, quantized transport of robust localized wave-packets. The integration of experimental topolectrical circuits confirms the practical feasibility of harnessing these effects, setting the stage for future explorations in larger, more complex lattices and alternative wave platforms such as photonic and acoustic metamaterials. These developments promise transformative applications in modern wave-based technologies, offering robust and tunable transport mechanisms operating beyond the linear regime.

As a culmination, the experimental realization of quantized soliton pumping via multiple Chern numbers reflects a profound understanding of nonlinear topological wave transport. This breakthrough bridges condensed matter theory, nonlinear dynamics, and applied physics, charting a course for innovations that leverage the robust topological nature of nonlinear excitations. The capacity to manipulate soliton trajectories with topological precision holds promise for scalable implementations in cutting-edge wave technologies and inspires theoretical pursuits in nonlinear topological phenomena.

Subject of Research: Quantized soliton pumping in nonlinear, time-modulated two-dimensional lattices governed by high-dimensional topological invariants including first and second Chern numbers.

Article Title: Quantized Soliton Pumping Governed by High-Dimensional Chern Numbers

Web References: DOI: 10.1093/nsr/nwag007

Image Credits: ©Science China Press

Keywords: soliton pumping, nonlinear lattices, topological transport, Chern numbers, topolectrical circuits, high-dimensional topology, fractional quantization, nonlinear dynamics, anisotropic transport, wave-packet localization, time-modulated lattices, topological invariants

Weyl semimetals have captivated intense research interest because of their unconventional topological properties, characterized by Weyl nodes—points in momentum space where conduction and valence bands touch—and their associated Fermi arc surface states, which defy classical surface state expectations. The team spearheaded an innovative approach by incorporating Rashba spin-orbit coupling into a time-reversal-symmetry-broken WSM model. This subtle addition induces nontrivial topological band structures culminating in a system characterized by a quantized Chern number of 1, a hallmark of quantum Hall states but now achieved in a 3D framework.

Deep theoretical modeling revealed the intricate band structures in both the bulk and surface states of the proposed system. The researchers demonstrated the emergence of unique boundary manifestations unlike those observed in conventional stacked 2D Chern insulators. Along one spatial direction, two distinct chiral surface states propagate unidirectionally, while along another axis, a pair of hinge states appear, their chirality decisively linked to the Fermi energy. Compellingly, these distinct topological states are interwoven by additional chiral surface states along the third spatial dimension, collectively embodying a novel 3D bulk-boundary correspondence principle.

One of the most striking findings is the anisotropic nature of electrical transport in this 3D QAHE phase. The Hall resistance does not assume a universal value but instead varies discretely depending on the current direction and precise Fermi energy placement. The resistance quantization takes values of 0, h/e², or ±h/e², revealing a rich landscape of transport regimes. These predictions were rigorously verified through Landauer-Büttiker formalism-based transport calculations, which also indicated remarkable resilience to typical disorder effects. Such robustness is critical for practical applications, as it signals the stability of the quantum state under realistic imperfections.

This multidimensional topology fundamentally distinguishes the system from mere layer stacking of 2D quantum anomalous Hall states, establishing a genuinely 3D quantum Hall insulator with complex interplay among surface and hinge modes. The implication of these findings is profound, suggesting that 3D topological phases can host exotic electronic phenomena inaccessible by conventional 2D systems, potentially enabling new paradigms in dissipationless transport and quantum computation.

The practical ramifications extend beyond academic curiosity. The ability to harness a stable 3D QAHE phase paves the way for a new class of low-power, topologically protected devices contributing to programmable electronics and in-memory computing architectures. These applications rely on the precise control of edge and surface states in 3D geometries, enabling robust, high-density, and energy-efficient information processing components well suited for the technological demands of the future.

To achieve experimental realization, the team points toward magnetically doped WSM compounds, which break time-reversal symmetry essential for the QAHE. Such materials are increasingly accessible due to advances in material synthesis and precision doping techniques. The experimental pursuit will likely focus on detecting quantized Hall resistance signatures and the distinctive anisotropic transport behaviors predicted, which serve as fingerprints of the 3D QAHE phase.

The understanding of 3D QAHE enriches the broader landscape of topological phases, exemplifying how spin-orbit coupling and magnetic order can intertwine to produce complex, emergent phenomena in quantum materials. This synergy enriches theoretical topological classification schemes and challenges experimentalists to explore emergent quasiparticles and boundary modes beyond conventional paradigms.

Moreover, this work underscores the critical importance of multidirectional chiral surface and hinge states in defining the electronic architecture of novel quantum phases. The ability to manipulate these states via Fermi energy tuning or directional current injection offers a tantalizing prospect for device-level control, paving the way for engineered topological circuits where information is encoded and transported with unprecedented fidelity.

In the broader context of condensed matter physics, this proposal marks a pivotal step in completing the Hall effect family, transitioning from 2D quantum anomalous Hall systems to fully fledged 3D analogs. Such progress not only satisfies longstanding theoretical quests but opens a new frontier in material functionalities endowed by topology, spin, and magnetic interactions.

While the theoretical promise is unequivocal, the path to experimental validation will require meticulous material design and measurement precision. Potential challenges include maintaining the delicate balance of magnetic doping, disorder management, and achieving the requisite Rashba spin-orbit coupling strength. Nonetheless, the roadmap provided by this study equips experimentalists with clear target parameters and transport signatures, accelerating the realization of these quantum states in laboratory settings.

Ultimately, the discovery of the three-dimensional quantum anomalous Hall effect in Weyl semimetals represents a transformative leap in the understanding and application of topological quantum materials. It exemplifies the power of theoretical innovation combined with deep physical insights, forging a path toward novel quantum devices that leverage the intricate dance of electrons in topologically nontrivial landscapes.

Subject of Research: Quantum anomalous Hall effect in three-dimensional Weyl semimetals

Article Title: Quantum Hall Effect Goes 3D: Scientists Unveil New Topological State in Weyl Semimetals

Web References: DOI: 10.1016/j.scib.2025.09.037

Image Credits: ©Science China Press

Keywords: Quantum anomalous Hall effect, 3D QAHE, Weyl semimetals, Rashba spin-orbit coupling, topological insulators, Chern number, chiral surface states, hinge states, anisotropic transport, Landauer-Büttiker calculations, magnetically doped materials, topological electronics