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.

We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic, periodically kicked, interacting, quantum many-body systems by analytically calculating the spectral form factor (SFF), $K(t)$, up to two leading orders in time, $t$. We explicitly consider the presence or absence of time-reversal ($\mathcal{T}$) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for $\mathcal{T}$-invariant systems with ${\mathcal{T}}^2=1$, the circular unitary ensemble (CUE) in the absence of $\mathcal{T}$ symmetry, and the circular symplectic ensemble (CSE) for $\mathcal{T}$-invariant systems with ${\mathcal{T}}^2=-1$. Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in $t$ for the COE and CUE classes is finite, the number of such diagrams is exponentially large in $t$ for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time $t^{*}$, the SFF takes the form $K(t)\simeq 2t-2t^2/\mathcal{N}$ up to second order in time, where $\mathcal{N}$ is the Hilbert space dimension. For the CUE class, we show that $K(t)\simeq t$ beyond $t^{*}$, and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that $K(t)\simeq 2t+2t^2/\mathcal{N}$ up to two orders in time beyond $t^{*}$. The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of $t^{*}$ with system size $L$ is determined by the eigenvalues of a doubly stochastic matrix $\mathcal{M}$. For strongly interacting fermionic chains, $\mathcal{M}$ is $SU(2)$ invariant in all three cases, leading to $t^{*}\propto L^2$ in the presence of $U(1)$ symmetry. In the absence of $U(1)$ symmetry, we find $t^{*}\propto L^0$, due to the gapped, nondegenerate, second-largest eigenvalue of $\mathcal{M}$, or $t^{*}\propto \ln (L)$, due to the gapped second-largest eigenvalue with degeneracy proportional to $L^{\zeta }$, $\zeta >0$. Our calculation of the SFF is plausible in higher space dimensions as well, where similar system-size scalings of $t^{*}$ can be obtained.

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.

Robust inference for stochastic dynamical systems is often hampered by sparse sampling and the absence of closed-form likelihoods. We introduce a Monte Carlo path-inference framework that leverages full-path statistics and bridge processes to deliver reliable parameter estimation and model selection from coarsely sampled time series, without requiring analytical solutions. Crucially, we couple mechanistic stochastic models with their inference procedures to quantify how experimental design—specifically, sampling frequency and dataset size—governs estimator precision and model distinguishability. This analysis reveals optimal sampling regimes and sharp, resolution-dependent limits beyond which competing models become empirically indistinguishable. We validate the approach across four disparate systems—trajectories of optically trapped particles, human microbiome dynamics, social-media topic mentions, and forest population time series—recovering parameters and identifying when inference is fundamentally constrained by measurement resolution, thereby clarifying ongoing debates about dominant noise sources in these systems. Together, these results establish path-based Monte Carlo as a practical, general tool for inference and model discrimination in complex systems and provide principled guidelines for designing measurements that maximize information under real-world constraints.

For four decades, statistical physics has been providing a framework to analyze neural networks. A long-standing question remained on its capacity to tackle deep-learning models capturing rich feature-learning effects, thus going beyond the narrow networks or kernel methods analyzed until now. We positively answer through the study of the supervised learning of a multilayer perceptron. Importantly, (i) its width scales as the input dimension, making it more prone to feature learning than ultrawide networks and more expressive than narrow ones or ones with fixed embedding layers, and (ii) we focus on the challenging interpolation regime where the number of trainable parameters and data are comparable, which forces the model to adapt to the task. We consider the matched teacher-student setting. Therefore, we provide the fundamental limits of learning random deep neural-network targets and identify the sufficient statistics describing what is learned by an optimally trained network as the data budget increases. A rich phenomenology emerges with various learning transitions. With enough data, optimal performance is attained through the model’s “specialization” toward the target, but it can be hard to reach for training algorithms, which get attracted by suboptimal solutions predicted by the theory. Specialization occurs inhomogeneously across layers, propagating from shallow toward deep ones but also across neurons in each layer. Furthermore, deeper targets are harder to learn. Despite its simplicity, the Bayes-optimal setting provides insights into how the depth, nonlinearity, and finite (proportional) width influence neural networks in the feature-learning regime that are potentially relevant in much more general settings.

Turbulence follows a few well-known organizational principles, rooted in conservation laws. One such principle states that a system conserving two sign-definite invariants self-organizes into large-scale structures. Ordinary three-dimensional turbulence does not fall within this paradigm but is profoundly altered when subject to rotation. In rotating turbulence, 3D inertial waves coexist alongside emergent two-dimensional structures, which tend to take the form of domain-scale flows called condensates. This interplay raises a fundamental question: Why and when are 2D flows sustained if only 3D waves are excited? We develop a quasilinear wave-kinetic theory to answer this question. We show that near-resonant interactions between 3D waves and a large-scale 2D flow impose an additional conservation law: Waves must conserve their helicity separately for each helicity sign. This emergent sign-definite invariant constrains the waves to transfer their energy to large-scale 2D motions, which maintains the latter in statistical steady state. We derive analytical expressions for the 3D-2D energy transfer as a function of rotation, Reynolds number, and domain geometry in a rotation-dominated regime and compare them with extensive numerical simulations of the rotating 3D Navier-Stokes equations. As rotation increases, the energy transfer from the waves to the 2D flow progressively vanishes as the two decouple, leading to a transition between distinct classes of turbulence: from 2D-dominated to 3D-dominated wave turbulence. Our theory shows that this gradual transition is caused by a depletion of modes satisfying the resonance conditions and exhibits good agreement with numerical simulations when the number of near-resonant modes is not too small. We discuss such limitations of our theory, as well as the validity range of its underlying assumptions. Together, these results suggest a mechanism underlying two-dimensionalization in rotating turbulence and, more broadly, illustrate how nonlinear systems sustaining waves can self-organize into anisotropic, zero-frequency structures.

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.