Bounding the correlations predicted by quantum theory is an important challenge in quantum information science. Today’s leading approach is semidefinite programming relaxations, but existing methods still cannot account for many relevant types of constraints. Here, we propose a general semidefinite relaxation methodology that can incorporate a breadth of constraints needed in various quantum correlation problems, thereby generalising the seminal Navascués-Pironio-Acín hierarchy. It yields useful results at reasonable computational cost. We showcase the methodology and its features by using it to address five different quantum information problems. These are (i) entanglement witnessing from imperfect measurement devices, (ii) certifying measurements from fidelity-constrained sources, (iii) computing dimensionality in genuine multiparticle entangled states, (iv) benchmarking dimensionality for state preparation devices, and (v) finding uncertainty relations for nearly anticommuting observables. These applications reflect both the usefulness and versatility of the methodology, as well as its potential for broader relevance in the field.

Long-term stability in materials is commonly associated with resistance to external perturbations. Yet metastable defect populations may relax spontaneously even in the absence of environmental stimuli, reshaping macroscopic properties over extended timescales. Here, we show the real-space visualization of intrinsic aging of $n$-type ${\mathrm{Mg}}_3{(\mathrm{Sb},\mathrm{Bi})}_2$ under inert and room-temperature conditions after two-year storage by atomic-scale characterizations. Density-functional-theory calculations and machine-learning molecular dynamics simulations prove that the aging originates from relaxation of a metastable Mg-rich state, in which low vacancy-formation energies and migration barriers enable thermodynamically favorable and kinetically accessible Mg redistribution. Multiscale characterizations further reveal that grain boundaries (GBs) act as fast-diffusion pathways and effective sinks, establishing a hierarchical Mg redistribution process from grain interiors to GBs and surfaces. These results identify intrinsic defect physics as one of the fundamental stability constraints in functional materials and underscore the importance of controlling defect thermodynamics and transport pathways for achieving practically durable materials.

In the study of electronic structure, the wave function view dominates the current research landscape and forms the theoretical foundation of modern quantum mechanics. In contrast, valence bond (VB) theory represents chemical bonds as shared electron pairs and can provide an intuitive, particle-based insight into chemical bonding. In this work, using a newly developed periodic dynamic Voronoi Metropolis sampling (PDVMS) method, we project classical many-body electronic configurations from the neural network wave function and apply VB theory to construct a complementary particle-view paradigm. The powerful neural network wave function can achieve near-exact ab initio solutions for the ground state of both molecular and solid systems. It allows us to definitively characterize the ground state of benzene by reassessing the competition between its two VB structures. Extending PDVMS to solids, we also predict a spin-staggered VB structure in graphene, which explains emergent magnetic properties in graphene-based nanostructures. Furthermore, this particle view provides insight into the optimization process of the neural network wave function itself. Our work thus introduces a framework for analyzing many-body electronic structure in molecules and solids, opening new avenues for investigating this complex problem and its associated exotic phenomena.

The generation of non-Gaussian quantum states is a key requirement for universal continuous-variable quantum information processing. We report the experimental generation of large-amplitude squeezed coherent-state superpositions (squeezed cat states) on free-space optical pulses, reaching an amplitude of $\alpha =2.47$, which, to our knowledge, exceeds all previously reported values. Our protocol relies on the controlled mixing of the Fock states $|1⟩$ and $|2⟩$ through a tunable beam splitter, followed by heralding via homodyne detection. The resulting state displays three well-resolved negative regions in its Wigner function and achieves a fidelity of 0.53 with the target state $\propto \widehat{S}(z)(|\alpha ⟩-|-\alpha ⟩)$, with $\alpha =2.47$ and squeezing parameter $z=0.56$. These results constitute a significant milestone for temporal breeding protocols and for the iterative generation of optical Gottesman-Kitaev-Preskill states, opening perspectives for scalable and fault-tolerant photonic quantum architectures.

Superionic conductors are fascinating systems representing a state of matter intermediate between solids and liquids, yet their underlying physics remain poorly understood. Correlations in jump dynamics have been theoretically proposed as a key mechanism underlying superionic diffusion, but a direct experimental demonstration remains elusive. Here, we report the direct observation of collective cation hopping in the argyrodite compound ${\mathrm{Cu}}_7{\mathrm{PS}}_6$, using single-crystal neutron scattering to access the full four-dimensional space-time correlations of ionic motion. By leveraging the rarely used coherent scattering channel in quasielastic neutron scattering, together with diffuse scattering, we reveal pronounced short-range dynamical correlations between diffusing Cu ions, providing direct experimental evidence of collective hopping in a superionic phase. Machine-learned molecular dynamics simulations quantitatively reproduce all our measurements and pinpoint the dominant collective hopping mechanism. Furthermore, we establish through both experiments and simulations that the collective diffusive hops originate from the breakdown of low-energy phonons dominated by the Cu motions in the parent ordered phase, demonstrating a key link between phonon precursors and the activation of facile collective hops. This work presents the first direct experimental evidence of collective hopping in superionic argyrodites and identifies wave-vector-resolved soft phonons as its physical origin, advancing the microscopic understanding of ionic transport in superionic solids.

Divalent atoms have emerged as powerful alternatives to alkalis in ultracold atom platforms, offering unique advantages arising from their two-electron structure. Among these species, ytterbium (Yb) is especially promising, yet its anionic properties and its Rydberg spectrum remain comparatively unexplored. In this work, we perform a comprehensive experimental and theoretical investigation of ultralong-range Rydberg molecules (ULRMs) of ${}^{174}\mathrm{Yb}$ in $6sns{{}^{1}S}_0$ Rydberg states across nearly two decades in principal quantum number $n$ and 3 orders of magnitude in molecular binding energy. Using the Coulomb Green’s function formalism, we compute Born-Oppenheimer molecular potentials describing the Rydberg atom in the presence of a ground-state perturber and achieve quantitative agreement with high-resolution molecular spectra. This enables the extraction of low-energy electron-Yb scattering phase shifts, including the zero-energy $s$-wave scattering length and the positions of two spin-orbit split $p$-wave shape resonances. Our results provide strong evidence that the ${\mathrm{Yb}}^{-}$ anion exists only as a metastable resonance. We additionally show the sensitivity of ULRM spectra to the atomic quantum defects, using this to determine the quantum defect of the $6s23f{{}^{1}F}_3$ state. Together, these findings establish Yb ULRMs as a powerful probe of electron-Yb interactions and lay essential groundwork for future Rydberg experiments with divalent atoms.

Ab initio auxiliary-field quantum Monte Carlo (AFQMC) is a systematically improvable many-body method, but its application to extended solids has been severely limited by unfavorable computational scaling and memory requirements that obstruct rigorous access to the thermodynamic and complete-basis-set limits. By combining tensor hypercontraction with $\mathbf{k}$-point symmetry, we reduce the computational and memory scaling of ab initio AFQMC for solids to $\mathcal{O}(N^3)$ and $\mathcal{O}(N^2)$, respectively, with an arbitrary basis, comparable to diffusion Monte Carlo. This enables simultaneous, controlled extrapolation of full-system AFQMC calculations to the thermodynamic and complete-basis-set limits across insulating, metallic, and strongly correlated solids, without embedding, local approximations, empirical finite-size corrections, or composite schemes. Our results establish AFQMC as a general-purpose, systematically improvable alternative to diffusion Monte Carlo and coupled-cluster methods for predictive ab initio simulations of solids, enabling accurate energies and magnetic observables within a unified framework.

The appearance of fractional Chern insulators in moiré systems can be rationalized by the presence of a fictitious magnetic field associated with the spatial texture of layer-resolved electronic wave functions. Here, we present a systematic study of real-space topology and the associated fictitious magnetic fields in moiré systems. We first show that at the level of individual Bloch wave functions, the real-space Chern number, akin to a Pontryagin index, is a fragile marker. It generically vanishes except for specific limits where the Bloch functions exhibit fine-tuned zeros within the unit cell, such as the chiral limit of twisted bilayer graphene (TBG) or the adiabatic regime of twisted homobilayer transition metal dichalcogenides (TMDs). We then show that these limitations do not apply to textures associated with ensembles of Bloch wave functions, such as entire bands or the ensemble of states at a given energy. The Chern number of these textures defines a robust topological index protected by a spectral gap. We find that symmetries constrain it to be nonzero for both twisted TMDs and TBG across all twist angles and levels of corrugation, which can be verified in scanning tunneling microscopy measurements. By projection to the band ensemble texture, a single-component Hamiltonian under fictitious magnetic field emerges in broad regime, enabling a direct comparison of multicomponent bands to Landau-level wave functions. We also study real-space topology within the topological heavy fermion model of TBG, finding that the real-space topological features are supported only by the light $c$ electrons.

The eigenstate thermalization hypothesis (ETH) posits how isolated quantum many-body systems thermalize, assuming that individual eigenstates at the same energy density have identical expectation values of local observables in the limit of large systems. While the ETH apparently holds across a wide range of interacting quantum systems, in this work, we show that it may require generalization in the presence of thermal first-order phase transitions. We introduce a class of all-to-all spin models, featuring first-order thermal phase transitions that stem from two distinct local maxima of entropy (two mean-field solutions that we dub “branches”) that exchange dominance in the many-body density of states as the energy is varied. We argue that, for energies in the vicinity of the thermal phase transition, eigenstate expectation values do not need to converge to the same thermal value. The system has a regime with coexistence of two classes of eigenstates corresponding to the two branches with distinct expectation values at the same energy density and another regime with Schrödinger-cat-like eigenstates that are interbranch superpositions; these two regimes are separated by an eigenstate phase transition. We propose a more general form of the ETH Ansatz, support our results by semiclassical calculations and an exact diagonalization study of a microscopic spin model, and argue that the structure of eigenstates in the vicinity of thermal first-order phase transitions can be experimentally probed via nonequilibrium dynamics.

Ferroelectric domain walls are described as interfacial regions between oppositely polarized domains, where the original bulk distortions are suppressed or reconstructed at energetic cost. Here, we demonstrate that $\mathrm{Pca}{2}_1$ ${\mathrm{HfO}}_2$ exhibits a distinct ferroelectric domain wall where the polarization-reversal boundaries remain structurally bulk compatible and exceptionally low in energy. The central mechanism is a pair-condensation—the simultaneous condensation of the two trilinearly coupled phonon modes located at opposite termini of a single phonon branch. We show that this pair condensation not only establishes the microscopic origin of layer-selective ferroelectricity in bulk but also governs the exceptionally low energy of the 180° domain walls. Specifically, the simultaneous reversal and complementary energetic responses of the paired condensed modes compensate the domain-wall energy cost and facilitates a wall-centered structure where bulk motifs are locally recovered rather than suppressed. This bulklike domain wall is corroborated by the agreement between density-functional-theory calculations and annular-bright-field-STEM measurements of oxygen displacements. This logic extends to suboptimal and higher-energy domain walls which emerge as accessible structural bridges via the selective, energy-minimizing reversal of individual condensed phonon modes. By unifying these structural and energetic observations under the principle of pair condensation, this work provides a generalized predictive framework for engineering domain-wall physics in multimode functional materials.

The origin of superconductivity in magic-angle twisted bilayer graphene has been a subject of intense debate. While some experimental evidence indicated an unconventional pairing mechanism which should be sensitive to Coulomb screening, experimental attempts to tune the critical temperature by screening Coulomb interactions so far have remained unsuccessful, possibly indicating a conventional phonon-mediated pairing. Here we study a double-layer electronic system consisting of two twisted graphene bilayers in immediate proximity of each other but remaining electronically decoupled. By increasing the carrier density in one bilayer, we completely suppressed both the superconductivity and the correlated-insulator state in the adjacent magic-angle graphene. The observation of such an effect from screening offers support for an unconventional mechanism of Cooper pairing in magic-angle twisted bilayer graphene, shedding new light on the underlying physics governing their properties.

We present a general, systematic, and quantitative theory of electromagnetism and chirality in crystalline solids. Symmetry is its basic guiding principle, enabling us to consider macroscopic multipole densities without reference to any specific microscopic configurations. We use a formal analogy between space inversion $i$ and time inversion $\theta$ to identify two complementary, comprehensive classifications of crystals, based on five categories of electric and magnetic multipole order—called polarizations—and five categories of chirality. The five categories of polarizations (parapolar, electropolar, magnetopolar, antimagnetopolar, and multipolar) embody the variety of ways in which electromagnetic multipole order can be realized in solids, thus expanding the familiar notion of electric dipolarization in ferroelectrics and magnetization in ferromagnets to higher-rank multipole densities characterizing, e.g., altermagnets and alterelectrics. Group theory also permits the absolute, gauge-invariant quantification of these multipole densities. The five categories of chirality (parachiral, electrochiral, magnetochiral, antimagnetochiral, and multichiral) extend the notion of enantiomorphism—conventionally associated with the lack of spatial mirror symmetries—to include all possibilities for creating nonsuperposable images by applying the inversions $i$, $\theta$, and $i\theta$. In particular, multichiral systems lack all inversion symmetries and therefore have four different enantiomorphs. Each category of chirality is shown to arise from particular superpositions of electric and magnetic multipole densities. Our theory has great predictive power by relating electric and magnetic multipolar order with, e.g., distinctive features in the crystal structure, the electronic band structure, property tensors, and response tensors of quantum materials. These findings are critical for a large community working on fundamental and applied aspects of novel quantum materials.

Spin space groups (SSGs) impose fundamentally different constraints on magnetic configurations in real and reciprocal spaces. As a consequence, the correspondence between real-space and momentum-space spin arrangements is far richer than traditionally assumed. Building on the complete enumeration of SSGs, we develop a systematic, symmetry-based framework that classifies all possible spin arrangements allowed by these groups. This unified approach naturally incorporates conventional magnetic orders, altermagnetism, and coplanar odd-wave magnetism (including the previously proposed $p$-wave case) as distinct symmetry classes. Moreover, by determining the symmetry-allowed wave patterns of spin-polarized phases, our framework classifies coplanar odd-wave magnets beyond the $p$-wave case into higher-wave types ($f$, $h$, $j$, and $l$ wave). Crucially, our classification predicts a variety of novel magnetic phases, highlighted by the discovery of the coplanar even-wave magnet(including $d$, $g$, and $i$ wave): A state that is noncollinear in real space but hosts a collinear even-wave spin polarization in $\mathbf{k}$ space. Analysis of a minimal model reveals that this phase is characterized by nonquantized spin polarization and exhibits a novel mechanism for symmetry-enforced zero polarization on nondegenerate bands. Extending the framework from bulk crystals to layer SSGs appropriate for two-dimensional systems, we further predict layered counterparts and provide symmetry guidelines for designing bilayer coplanar odd-wave and even-wave magnets. We further validate this finding through first-principles calculations and propose ${\mathrm{CoCrO}}_4$ as a promising candidate for its experimental realization, thereby demonstrating the completeness and predictive power of the SSG-based classification of magnetic orders.

Analog quantum simulators offer a route to exploring strongly correlated many-body dynamics beyond classical computation, but their predictive power remains limited by the absence of quantitative error estimation. Establishing rigorous uncertainty bounds is essential for elevating such devices from qualitative demonstrations to quantitative scientific tools. Here we introduce a general framework for bounded-error quantum simulation, which provides predictions for many-body observables with experimentally quantifiable uncertainties. The approach combines Hamiltonian and Lindbladian Learning—a statistically rigorous inference of the coherent and dissipative generators governing the dynamics—with the propagation of their uncertainties into the simulated observables, yielding confidence bounds directly derived from experimental data. We demonstrate this framework on trapped-ion quantum simulators implementing long-range Ising interactions with up to 51 ions. We analyze error bounds on two levels. First, we learn an open-system model from experimental data collected in an initial time window of quench dynamics, simulate the corresponding master equation, and quantitatively verify consistency between theoretical predictions and measured dynamics at long times. Second, we discuss short-time error bounds directly from experimental measurements alone, without relying on classical simulation—crucial for entering regimes of quantum advantage. In both cases, the learned models reproduce the experimental evolution within the predicted bounds, demonstrating quantitative reliability and internal consistency. By integrating statistical learning, open-system modeling, and precise experimental control, bounded-error quantum simulation provides a scalable foundation for trusted analog quantum computation, bridging the gap between experimental quantum platforms and predictive many-body physics. The techniques presented here directly extend to digital quantum simulation.

We investigate how polycrystallinity evolves in homogeneously nucleated crystals in an experimental hard-sphere-like colloidal system using three-dimensional confocal microscopy. We capture the whole process of nucleation and crystal growth and use a recently developed method to resolve grain structure and misorientation between adjacent grains and nearest-neighbor clusters. Closely following nucleation, crystals are shown to give rise to grains through the emergence of stacking faults and twinning. The birth rate of new grains from a single nucleation event are measured, observed to increase, and then plateau. We visualize the spatial structure of a nucleus which comprises a highly ordered core and a radial increase in misorientation angles. Finally, we use local misorientation to formulate a measure for positional order which reveals a sharp decoupling between the development of orientational order and positional order at times significantly after nucleation.

P-type MgAgSb with excellent room-temperature thermoelectric performance holds great promise for cooling and power generation. However, its practical application has been significantly limited by phase transitions, which are long considered a polymorphic transformation where the compositions remain identical but the crystal structures differ. Here we report the non-polymorphic nature of the complex phase transitions in Mg-Ag-Sb during thermal cycling, yielding four compositionally distinct ternary compounds that crystallize in three different structures. During heating, the tetragonal $\alpha$-phase MgAgSb decomposes into the tetragonal $\beta$-phase ${\mathrm{Mg}}_5{\mathrm{Ag}}_{4.32}{\mathrm{Sb}}_{4.5}$, and subsequently into the cubic ${\gamma }_{\mathrm{H}}$-phase ${\mathrm{Mg}}_5{\mathrm{Ag}}_{2.11}{\mathrm{Sb}}_4$, with the progressive precipitation of ${\mathrm{Ag}}_3\mathrm{Sb}$ and Sb phases. Upon cooling, the ${\gamma }_{\mathrm{H}}$-phase ${\mathrm{Mg}}_5{\mathrm{Ag}}_{2.11}{\mathrm{Sb}}_4$ converts to the cubic ${\gamma }_{\mathrm{L}}$-phase ${\mathrm{Mg}}_5{\mathrm{Ag}}_{1.18}{\mathrm{Sb}}_{3.63}$, accompanied by the $\alpha$ phase with substantial ${\mathrm{Ag}}_3\mathrm{Sb}$ and Sb phases. The electron localization function and bonding analysis indicate that the Mg-Ag-Sb compound is characterized by ionic interactions, wherein the Ag-Sb bond is notably weaker compared to the Mg-Sb bond. Molecular dynamics simulations reveal that significant Mg and Ag migration within the Sb sublattices underlies the phase evolution. Surprisingly, the rapid atomic diffusion also enables the restoration of the $\alpha$-phase MgAgSb upon annealing at low temperatures. Successful recovery of the $\alpha$ phase can be realized even after 500 thermal cycles between 423–673 K, demonstrating unprecedented thermal recoverability.

Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a simulator of the Hubbard model with nearest-neighbor hoppings [L. H. Kendrick et al., arXiv:2509.18075.] showed that the isothermal compressibility $\kappa (\delta )$ has a maximum as a function of doping $\delta$. Here, we use the two-particle self-consistent plus approach to explain these experiments and connect the maximum in $\kappa (\delta )$ to the transformation of the single-particle spectrum from the pseudogapped to the metallic regime. This elucidates the nature of the pseudogap. Specifically, the maximum in $\kappa (\delta )$ practically coincides with the doping at which the precursor of the lower $(\pi ,\pi )$ spin-density-wave (SDW) band at the antinodal point crosses the zero-frequency $\omega =0$. The Knight shift ${\chi }_{\mathrm{sp}}(0,0)(\delta )$, as a function of doping, should also have a maximum. In addition, we demonstrate that two-particle self-consistent plus correctly predicts a maximum in the temperature dependence of the Knight shift ${\chi }_{\mathrm{sp}}(0,0)(T)$, consistent with recent ultracold-atom experiments [T. Chalopin et al., Proc. Natl. Acad. Sci. U.S.A. 123, e2525539123 (2026).]. The maxima in both quantities should exist, at sufficiently low temperatures ($T$), in both the intermediate $U\approx U_{\text{Mott}}$ and weak $U<U_{\text{Mott}}$ interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at ${\delta }_{\max }(T)$ can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at $\delta ={\delta }_{\max }$. We predict that, at low $T$, the multiple peaks in the spin susceptibility in the incommensurate case lead to more than two SDW precursor peaks in the spectral function and density of states. By allowing access to parameter regimes relevant to cuprates—including further-neighbor hopping ($t^{\prime },t^{\prime \prime }$) and low temperatures, our work provides a high-impact tool for further studies by the broader community.

The ability to control and manipulate time-reversal ($T$) symmetry-breaking phases with near-zero net magnetization is a sought-after goal in spintronic devices. The recently discovered hexagonal altermagnet manganese telluride ($\alpha$-MnTe) is a prime example. It has a compensated altermagnetic ground state where the magnetic moments are aligned in each layer and stacked antiparallel along the $c$ axis, yet it exhibits a spontaneous anomalous Hall effect (AHE) that breaks the $T$-symmetry with a vanishingly small $c$-axis ferromagnetic moment. However, the presence of three 120° separated in-plane magnetic domains presents a challenge in understanding the origin of the AHE and the effective control of the altermagnetic state. Here we use neutron scattering to show that symmetry breaking anisotropic strain, induced by compressive uniaxial pressure along the nearest-neighbor (NN) Mn-Mn bond directions, detwins $\alpha$-MnTe into a single in-plane magnetic domain. This control over in-plane domains allows us to unambiguously establish that the in-plane moments are aligned along the next-nearest-neighbor (NNN) Mn-Mn bond direction, irrespective of the applied strain directions. Mounting the sample on a piezoelectric strain cell along both NN and NNN directions can drive the sample into a single-domain state that significantly sharpens the AHE hysteresis loop and extends the AHE to lower temperatures. Furthermore, tuning the uniaxial strain reverses the sign of the AHE near room temperature. Remarkably, this is achieved without altering the altermagnetic phase-transition temperature. Combined with our phenomenological model, we propose that these effects result from the modification of the electronic Berry curvature by a combination of both spin-orbit coupling and strain. Our work not only unambiguously establishes the relationship between the in-plane moment direction and the AHE in $\alpha$-MnTe but also paves the way for future applications in highly scalable, strain-tunable magnetic sensors and spintronic devices.

Rigidity percolation is a crucial framework for describing rigidity transitions in amorphous systems. We present an efficient algorithm to study central-force rigidity percolation in two dimensions. This algorithm combines the pebble game algorithm, the Newman-Ziff approach to connectivity percolation, as well as novel rigorous results in rigidity theory, to exactly identify rigid clusters over the full bond concentration range, in a time that scales as $N^{1.02}$ for a system of $N$ nodes. We perform extensive numerical simulations with systems larger than 500 million nodes, far beyond the previous limitations. We obtain precise estimates for the critical exponents $\nu =1.1694(8)$ and $D_f=1.8423(7)$ and locate the critical threshold at $p_c=0.6602741(4)$. Besides opening the way to further accurate numerical studies of rigidity percolation, our work provides rigorous theoretical insights on specific cluster merging mechanisms that distinguish it from the standard connectivity percolation problem.

The discovery of nickelate superconductors has opened a new arena for studying the behavior of correlated electron liquids that give rise to unconventional superconductivity. While critical information about a material’s charge dynamics is encoded in its plasmons, collective modes of the electron gas, these excitations have not yet been observed in nickelate materials. Here, we use resonant inelastic x-ray scattering to detect plasmons in the metallic, low-valence nickelate ${\mathrm{Pr}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. Although qualitatively similar to those in cuprates, the nickelate plasmons are more heavily damped and have a lower velocity than those in a cuprate at comparable doping, which we attribute to reduced electronic hopping and enhanced screening of the long-range Coulomb interactions. Furthermore, the plasmons in ${\mathrm{Pr}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ soften with increasing temperature, in contrast to the cuprate, where plasmons remain at nearly fixed energy but become more strongly damped. Taken together, these results reveal a distinct charge-screening landscape in nickelates and place quantitative constraints on analogies to cuprates.