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.

We construct a family of static, geometrically local Hamiltonians that inherit eigenstate properties of periodically driven (Floquet) systems. Our construction is a variation of the Feynman-Kitaev clock—a well-known mapping between quantum circuits and local Hamiltonians—where the clock register is given periodic boundary conditions. Assuming the eigenstate thermalization hypothesis (ETH) holds for the input circuit, our construction yields Hamiltonians whose eigenstates have properties characteristic of infinite temperature, like volume-law entanglement entropy, across the whole spectrum—including the ground state. We then construct a family of exactly solvable Floquet quantum circuits whose eigenstates are shown to obey the ETH at infinite temperature. Combining the two constructions yields a new family of local Hamiltonians with provably volume-law entangled ground states, and the first such construction where the volume law holds for all contiguous subsystems.

Mechanical energy accelerates many physicochemical processes, including materials syntheses that are hard to produce with thermal energy alone. However, physical understanding connecting applied mechanical forces with internal stresses and ensuing reaction mechanisms is lacking. Here we demonstrate mechanical force-enabled synthesis and nanoscale patterning to metallize a two-dimensional (2D) material, producing an atomically thin superconductor. Localized force applied by atomic force microscope tips to van der Waals (vdW) encapsulated stacks of 2D bilayer ${\mathrm{MoTe}}_2$ and adjacent source Pd guides 2D ${\mathrm{Pd}}_7{\mathrm{MoTe}}_2$ growth with 50 nm lateral resolution. Force accelerates reaction kinetics exponentially per Eyring’s stress-assisted thermal activation model, reducing synthesis temperatures from $\sim 200°\mathrm{C}$ to near-room temperature. Finite element simulations, density functional theory, and ab initio grand canonical Monte Carlo calculations show that tip-induced compression facilitates Pd chemisorption to tensile-strained ${\mathrm{MoTe}}_2$ that then converts to uniform ${\mathrm{Pd}}_7{\mathrm{MoTe}}_2$. This demonstrates a new, generalizable paradigm for nanoscale synthesis of quantum materials, and high-precision engineering of superconductivity.

Quantifying physical concepts in terms of the ultimate performance of a given task has been central to theoretical progress, as illustrated by thermodynamic entropy and entanglement entropy, which quantify irreversibility and quantum correlations, respectively. Symmetry breaking is equally universal yet lacks such an operational quantification. While an operational characterization of symmetry breaking through asymptotic state-conversion efficiency is a central goal of the resource theory of asymmetry (RTA), such a characterization has so far been completed only for the U(1) group among continuous symmetries. Here, we identify the complete measure of symmetry breaking for a general continuous symmetry described by any compact Lie group. Specifically, we show that the asymptotic conversion rate between many copies of pure states in RTA is determined by the quantum geometric tensor, thereby establishing it as the complete measure of symmetry breaking. As an immediate consequence of our conversion rate formula, we also resolve the Marvian-Spekkens conjecture on conditions for reversible conversion in RTA, which has remained unproven for over a decade. Leveraging the connection between symmetry breaking and the theory of quantum reference frames, we also systematically introduce a standardized reference state for frameness based on our asymptotic conversion theory. In addition, by applying our analysis to a standard quantum-thermodynamic scenario, we show that asymptotic state conversion in contact with heat baths generally requires macroscopic coherence in the thermodynamic limit.

Recent experimental and theoretical developments in many-body quantum systems motivate the study of their out-of-equilibrium properties through multitime correlation functions. We consider the dynamics of higher-order out-of-time-order correlators (OTOCs) in a minimal circuit model for quantum dynamics. This model mimics the dynamics of a structured subsystem locally coupled to a maximally random environment. We prove the exponential decay of all higher-order OTOCs and fully characterize the relevant timescales, showing how local operators approach free independence at late times. We show that the effects of the environment on the local subsystem can be captured in a higher-order influence matrix, which allows for a Markovian description of the dynamics provided an auxiliary degree of freedom is introduced. This degree of freedom directly yields a dynamical picture for the OTOCs in terms of free cumulants from free probability, consistent with recent predictions from the full eigenstate thermalization hypothesis. This approach and the relevant influence matrix are expected to be applicable in more general settings and present a first step to characterizing quantum memory in higher-order OTOCs.

We propose a new classification of magnetic textures from a differential-geometric viewpoint. Magnetic textures are conventionally classified into collinear, coplanar, and noncoplanar magnets according to the relative arrangement of spins. These classes are typically characterized by the vector spin chirality (VSC) and the scalar spin chirality (SSC), which serve as indicators of noncollinearity and noncoplanarity, respectively. However, this conventional classification is incomplete: in particular, noncoplanar textures cannot be fully characterized by the SSC alone, as exemplified by conical magnets. To refine this classification, we analyze the curves and surfaces traced by spins in real space using differential geometry and introduce two novel scalar spin chiralities that properly characterize noncoplanarity: the geodesic scalar spin chirality and the torsional scalar spin chirality. These quantities are directly connected to the differential geometry of curves on surfaces: the former is associated with the geodesic curvature, which quantifies the deviation from a geodesic, while the latter is related to the torsion. Based on these chiralities, we identify three distinct classes of noncoplanar magnetic textures. Furthermore, in close analogy with the roles of the VSC and the conventional SSC in emergent electrodynamics, the geodesic SSC gives rise to novel emergent phenomena. By constructing a semiclassical theory that incorporates nonadiabatic effects and higher-order gradients of the spatial modulation of spins, we demonstrate that the geodesic SSC induces an emergent band asymmetry, leading to a nonreciprocal response as a quantum geometric effect. This mechanism is a purely orbital effect, requiring no spin-orbit coupling, and the resulting discussion runs in parallel with the conventional picture of the topological Hall effect driven by the SSC. The geometric viewpoint developed here will provide broad new insights into classification, quantum geometry, emergent electrodynamics, and a wider variety of emergent phenomena.

Many networks in nature and applications have an approximate low-rank structure in the sense that their connectivity structure is dominated by a few dimensions. It is natural to expect that dynamics on such networks would also be low-dimensional. Indeed, theoretical results show that low-rank networks produce low-dimensional dynamics whenever the network is isolated from external perturbations or input. However, networks in nature are rarely isolated. Here, we study the dimensionality of dynamics in recurrent networks with low-dimensional structure driven by high-dimensional inputs or perturbations. We find that dynamics in such networks can be high- or low-dimensional, and we derive mathematical conditions on the network structure under which linearized dynamics are high-dimensional. In many low-rank networks, dynamics are suppressed in directions aligned with the network’s low-rank structure, a phenomenon we term “low-rank suppression.” We show that several low-rank network structures arising in nature satisfy the conditions for generating high-dimensional dynamics and low-rank suppression. Our results clarify important but counterintuitive relationships between a recurrent network’s connectivity structure and the structure of its response to external input.

Mechanical metamaterials achieve their extraordinary properties through their intricate architectures. While amorphous designs can minimize directional bias relative to their regular counterparts, their vast configuration space poses a significant challenge for conventional design strategies. Here, we introduce a physics-constrained, energy-based model framework to navigate this complexity with machine learning. We formulate a multiobjective energy function that encodes desired macroscopic properties—specifically, a target negative Poisson ratio and approximate isotropy—into the network’s topology. The design problem is then recasted as finding the ground state of this energy landscape, while operating within a mechanically stable or physics-constrained configuration space. We employ Boltzmann annealing as a physically consistent inference algorithm to identify the optimal low-energy configurations. The structures discovered and fabricated via 3D printing exhibit a highly negative Poisson ratio together with near-equal responses along principal axes (i.e., isotropy along both $x$ and $y$ axes). Remarkably, these optimized structures also reveal highly desirable emergent properties, including exceptional performance in specific energy absorption, impact resistance, and fracture toughness, significantly outperforming regular lattice counterparts. This work establishes a robust and interpretable machine-learning framework for the design of high-performance amorphous metamaterials.

The sequential response of frustrated materials—ranging from crumpled sheets and amorphous media to metamaterials—reveals their memory effects and emergent computational potential. Despite their spatial extension, most studies rely on a single global stimulus, such as compression, effectively reducing the problem to scalar driving. Here, we introduce vectorial driving of frustrated materials by applying multiple spatially localized stimuli to explore path-dependent, sequential responses. We uncover a wealth of phenomena absent in scalar driving, including non-Abelian responses, mixed-mode behavior, and chiral loop transients. We show that fold singularities connect three states—ancestor, descendant, and sibling. This recurring pattern serves as the elementary building block of all sequential paths. We then introduce three levels of description of sequential, path-dependent responses. At the most fundamental level, path-dependent transition graphs (pt-graphs) and strain maps capture the response under arbitrary vectorial driving and connect pathways to the underlying singularities. They provide a complete description analogous to transition graphs (t-graphs) for scalar driving. However, as pt-graphs and strain maps become unwieldy for high-dimensional driving, we introduce b-graphs—graphs whose nodes and transitions encode the system response to binarized vectorial driving. These present a less complete but much simpler second-level description by restricting attention to binary input and their induced transitions. The resulting graphs form subgraphs of the pt-graph and naturally connect to sequential computing and Boolean circuits. We show that a single sample can encode multiple b-graphs that are selectable through the driving strength and can be reprogrammed via additional inputs. Finally, we introduce graph-based motifs that enable a systematic analysis of b-graphs and provide a practical tool for characterizing the statistical response of very high-dimensional systems, where a full pt- or b-graph description is neither feasible nor particularly insightful. As statistical measures of pathway complexity, these motifs can be obtained in systems of any size or complexity. Our work paves the way for strategies to explore, harness, and understand complex materials and memory while advancing embodied intelligence and in materia computing.

The most efficient and most versatile technique for producing highly charged ions (HCI) with high atomic numbers is electron stripping in-flight at high kinetic energies. While this requires relativistic velocities, many experiments call for trapped or at least slowly moving ions. For example, studies of fundamental interactions would significantly improve in precision when performed in an ion trap while studies on slow HCI for material science are not yet feasible due to the lack of availability. The goal of the HITRAP project at the GSI Helmholtzzentrum für Schwerionenforschung GmbH is to decelerate accelerator-produced heavy HCI and prepare them in a Penning trap via electron cooling for subsequent experiments. We present the trapping of accelerator-produced HCI. Fully stripped ${{}^{36}\mathrm{Ar}}^{18+}$ were decelerated from the production energy of $45\mathrm{MeV}/\mathrm{u}$ to $6\mathrm{keV}/\mathrm{u}$ followed by trapping in the HITRAP Cooling Trap for the duration of several seconds. The Cooling Trap is a nested Penning trap that allows for the simultaneous storage of HCI ions with a cold electron plasma. It was commissioned with HCI from a local electron beam ion source, which led to the first observation of electron cooling of highly charged ions in a Penning trap, marking a significant advancement in the field. These developments constitute a major step toward a worldwide unique facility with unparalleled capabilities in heavy-ion research and already enable the first experimental studies at HITRAP.

Here, we demonstrate that the time-evolving interface during droplet formation—and, in particular, the morphology near pinch-off—encodes sufficient physical information for machine-learning (ML) models to accurately infer key fluid properties, including viscosity and surface tension. A dataset was constructed from high-speed imaging of droplet formation in both dripping and and jetting regimes. The frame closest to breakup was selected and the contour of the droplet extracted and recorded, along with the corresponding flow conditions and fluid properties. Experiments were conducted using Newtonian fluids, such as silicon oils, aqueous solutions of ethanol and glycerin, and methanol, under controlled conditions, spanning $0.001<\mathrm{Re}<200$ and $0.01<\mathrm{Oh}<20$, providing 840 samples. Supervised regression models were trained to predict fluid properties from these inputs. The models accurately infer surface tension and viscosity across a wide range of fluids using the single high-speed snapshot. Moreover, apart from predicting fluid properties from a snapshot, the proposed framework can carry out the inverse problem and predict droplet shape at breakup based on fluid properties and flow conditions. Unsupervised clustering of the learned representations reveals distinct regions in the Re–Oh and Bo–Oh parameter spaces, indicating that the latent space captures meaningful physical structure and provides insight into droplet dynamics. These results establish a data-driven alternative to conventional measurement techniques, reducing experimental complexity and evaluation time while enabling integration into automated systems.

In open quantum systems hosting excitons, dissipation mechanisms critically shape the excitonic dynamics, band structure, and topological properties. A microscopic understanding of excitons in such non-Hermitian settings demands a first-principles generalization of the Bethe-Salpeter equation (BSE). Building on a recently introduced nonequilibrium Green’s function formalism compatible with Lindbladian dynamics, we derive a non-Hermitian BSE from diagrammatic perturbation theory on the Keldysh contour, and obtain a microscopic excitonic Hamiltonian that incorporates dissipation while preserving causality. We apply the formalism to valley excitons in transition metal dichalcogenides coupled to structured photon baths. We uncover a rich landscape of exceptional points in momentum space, forming either discrete sets or continuous manifolds, depending on bath structure. The exceptional points give rise to nonanalytic valley polarization, unusual polarization pattern in photoluminescence, and nontrivial topological signatures. Our results establish a first-principles framework for predicting and controlling excitonic behavior in open quantum materials, showing how engineered environments can be leveraged to induce and manipulate non-Hermitian and topological properties.