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.

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.

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.

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.

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.

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.

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.

Quantum sensors based on individual spins provide unprecedented access to local magnetic fields in condensed matter, chemistry, and biology, with solid-state defect spins emerging as the leading platform. However, their molecular-sensing capabilities are limited by confinement to a host lattice, which prevents placement in close proximity to a target molecule. Molecular spins offer an alternative, enabling chemical tunability and flexible positioning relative to the target system. Here we present a nanoscale sensing platform that combines molecular electron spins, ultrasensitive mechanical readout, and Hamiltonian engineering. Using a modified XYXY dipolar decoupling sequence, we suppress electron-electron dipolar interactions across a broad distribution of control fields, extending coherence times to $\sim 400\mathrm{\mu }\mathrm{s}$ in an attoliter-scale droplet containing $\sim 100$ trityl-OX063 radicals. Leveraging this sequence, we demonstrate frequency-selective detection of nanotesla-scale ac fields and perform sensing and spectroscopy of small, local nuclear-spin ensembles. Collectively, these results establish spin-based quantum integrated nanomechanical transduction as a framework for quantum sensing that affords molecular-level control over sensor properties and enables direct integration into complex molecular targets.

Discovering materials that combine topological phenomena with correlated electron behavior is a central pursuit in quantum materials research. Monolayer ${\mathrm{TaIrTe}}_4$ has recently emerged as a promising platform in this context, hosting robust quantum spin Hall insulator (QSHI) phases both within a single-particle gap and within a correlation-induced gap arising from van Hove singularities (vHSs), accessed via electrostatic doping. Its intrinsic monolayer nature offers exceptional tunability and the potential to realize a rich variety of interaction-driven topological phases. In this work, we combine theory and experiment to map the phase landscape of monolayer ${\mathrm{TaIrTe}}_4$. Using Hartree-Fock calculations, we investigate the interaction-driven phase diagram near the vHSs under commensurate filling conditions. By systematically tuning the dielectric screening and strain, we uncover a rich set of ground states—including QSHI, trivial insulator, higher-order topological insulator, and metallic phase—among which are interaction-driven topological phase transitions. Experimentally, we perform local and nonlocal transport measurements across a broad set of devices. Because of unavoidable strain variations during fabrication, the devices exhibit several distinct transport regimes, whose evolution is consistent with the theoretically predicted phase diagram. Finally, band-projected exact diagonalization together with insulating transport features at fractional fillings provide preliminary signatures consistent with a time-reversal-symmetric fractional QSHI in this system. Together, our results lay the groundwork for understanding correlation-driven topological phenomena in ${\mathrm{TaIrTe}}_4$ and open new directions for engineering exotic quantum phases in low-dimensional materials beyond the limitations of moiré superlattices.

Beyond the quasi-two-dimensional (2D) paradigm of cuprates, the role of the third dimension of the Ruddlesden-Popper bilayer nickelates is essential to decoding their superconducting mechanism. Here, using angle-resolved photoemission spectroscopy (ARPES) with varied photon energies, we systematically investigate the electronic band structures in three dimensions for superconducting ${(\mathrm{La},\mathrm{Pr},\mathrm{Sm})}_3{\mathrm{Ni}}_2{\mathrm{O}}_7/{\mathrm{SrLaAlO}}_4$ thin films (superconducting onset temperature $T_{\mathrm{c}}^{\mathrm{onset}}\sim 48\mathrm{K}$) transferred via a cryogenic ultrahigh vacuum (UHV) suitcase. We reveal an orbital-dependent dimensionality: While the $d_{{\mathrm{x}}^2\text{−}{\mathrm{y}}^2}$-dominant bands exhibit a quasi-2D character, the $d_{{\mathrm{z}}^2}$-dominant band displays a finite $k_{\mathrm{z}}$ dispersion. Finite energy gaps are identified on all observed bands across multiple high-symmetry directions. Systematic temperature-dependent analysis characterizes the superconducting nature of the gap on the $d_{{\mathrm{z}}^2}$-derived band, revealing a large gap $\mathrm{\Delta }\sim 18\mathrm{meV}$ and a ratio $2\mathrm{\Delta }/k_{\mathrm{B}}{T}_{\mathrm{c}}\sim 8$ exceeding the weak-coupling BCS limit. The suppression of spectral weight near the Fermi level persists above the superconducting transition temperature. Ubiquitous vertical spectral features evidence the presence of electron interactions. These results underscore the role of the $d_{{\mathrm{z}}^2}$ orbital and correlations, placing constraints on theoretical models for nickelate superconductivity.

In nonlinear transport, the quantum-geometric effects can generate higher-harmonic voltages in response to a driving current, which has defined a fast-moving field of intense interest. However, in realistic materials where disorder scattering also contributes to nonlinear transport, identifying the geometric mechanisms remains a challenge. In particular, a theoretical framework for data analysis is still lacking for nonlinear transport at any order. Here, we develop a mechanism-resolved and symmetry-guided framework for identifying mechanisms of third-order nonlinear transport in disordered materials. We find a total of 20 mechanisms of third-order nonlinear transport, by treating quantum-geometric and disorder-mediated mechanisms on an equal footing. More importantly, we propose a protocol of data analysis that combines symmetry diagnosis of magnetic point groups and scaling law of relation between the third-order nonlinear Hall conductivity and linear longitudinal conductivity. We identify characteristic fingerprints in the scaling-law weights, which allow the mechanisms to be quantitatively distinguished in experiments. We have applied the protocol to identify the geometric mechanisms in materials with and without time-reversal symmetry, including 2D materials, topological materials, and altermagnets. The theory can be generalized to arbitrary orders of nonlinear transport, further promoting nonlinear transport as a probe of geometric effects and phase transitions in quantum materials.