Papers

We determine the full persistence probability distribution for a non-Markovian stochastic process, motivated by first-passage questions arising in interacting spin systems and allied systems. We show that this distribution is governed by a distinguished Painlevé VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, $K_{\mathrm{sech}}=1/(2 \pi \cosh[(x-y)/2])$. The universal persistence exponent originally obtained by Derrida, Hakim and Pasquier is recovered as an asymptotic observable and acquires a natural geometric interpretation. In the stationary scaling regime, the persistence probability admits an exact Pfaffian decomposition into even and odd Fredholm determinants of the integrable sech kernel. These determinants are controlled by a unique global solution of a second-order nonlinear ordinary differential equation, which is identified as a particular Painlevé VI equation. The corresponding Painlevé VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlevé VI system governing persistence admits a direct geometric interpretation: the relevant solution coincides with the mean curvature of a one-parameter family of Bonnet surfaces immersed in $\mathbb R^3$. A folding transformation between such surfaces singles out the Painlevé VI equation with Manin coefficients $[0,0,0,0]$, which in particular governs the universal persistence distribution in the symmetric Ising case. In this framework, the persistence exponent is identified with the asymptotic mean curvature of the associated surface.

Proliferating cell collectives often develop an active growth layer near their boundary that regulates expansion and morphology, as observed in systems ranging from bacterial biofilms to epithelial tissues and tumor spheroids. While such layers have been attributed to diverse mechanisms, their microscopic origin remains unclear in many situations. Here, we show that micromechanical feedback alone provides a minimal mechanism for their emergence. We introduce a particle-based model of non-motile proliferating cells in which growth is locally inhibited by compressive stress, coupling division to mechanical interactions and generating an active growth layer without biochemical regulation. An emergent mechanical length scale, denoted by $\chi$, sets the extent of the proliferative region and controls the system's behavior across scales, governing growth dynamics, morphology and organizing internal stress and velocity fields. Coarse-graining the model yields a continuum description with no adjustable parameters, providing a microscopic foundation for existing approaches. When the colony expands into a passive environment, we observe and characterize fingering instabilities driven purely by mechanical feedback. These instabilities can be tuned through the system geometry relative to $\chi$, and leads to an exponential acceleration of colony growth, enhancing the collective growth rate. We further establish a correspondence with nutrient-depletion models, providing a route to study the statistical properties of expanding fronts within a minimal microscopic framework.

Packing under confinement could generate rich ordered structures through entropic effects, which is a fundamental problem in condensed matter, biophysics and material science. The influence of confinement to the anisotropic hard particles–particularly regarding the emergence of topological defect structures–remains poorly understood. Recent studies have shown that granular rods confined within circular boundaries can cluster into square like super-particles, forming four disclinations. In this study, we employ Monte Carlo simulations in the NPT ensemble to investigate how circular confinement influences the ordered structures of rounded-corner hard-squares with varying roundness. At low roundness, the system forms an integrated cross-shaped domain with tetratic order and four +1/4 disclinations in the corners, along with some column shifts. As roundness increases, we found a new partition structure, where particles self-assemble into six domains separated by six +1/4 disclinations and a central -1/2 disclination. Our findings reveal that the interplay between confinement geometry and colloid shape can drive entropy governed structural transitions, offering new insights for the design of topological metamaterials.

This book provides an introduction to path integral methods and their application to modeling atomistic processes. The book covers both the foundational theory and recently developed simulation techniques. The text provides a self-contained resource and was originally developed for the CECAM schools on Path Integral Methods.

We study the evolution of symmetry-resolved entanglement entropy in bulk-driven Floquet conformal field theories (CFTs). Focusing on the two-dimensional free compact boson CFT, we analyze how symmetry-resolved Rényi entropies approach or depart from equipartition among charge sectors. We show that the existence of an $\mathfrak{sl}^{(k)}(2,\mathbb{R})$ subalgebra of the Virasoro algebra introduces a free parameter, the label $k$, which allows us to control the breakdown of equipartition. We argue that this effect originates from an explicit coupling between low- and high-frequency modes. Based on a general oscillator representation of the Virasoro algebra, we expect this mechanism to persist beyond the free boson CFT. Finally, we discuss how the real-time dynamics of fine-grained symmetry-resolved entropies of a boundary state are modified under non-unitary evolution, which can be associated with post-selected weak measurements.

We present a theory of polarized photoluminescence of triplet excitons in semiconductor nanocrystal ensembles with the random fine structure contributed by the electron-hole exchange and carrier-nuclear hyperfine interactions. The interaction parameters are assumed to be normally and isotropically distributed. In particular, the exchange interaction is described by the Gaussian orthogonal ensemble of random matrices. The intensity of luminescence as well as the optical orientation and alignment are calculated as functions of the fine structure splitting parameters and the exciton lifetime. We have also analyzed the suppression of optical alignment and enhancement of optical orientation in an external longitudinal magnetic field.

We conducted replica exchange Monte Carlo simulations to investigate the phase diagram of identical hard rhombi systems in two dimensions. The rhombi shape varies from nearly square-like, as their minor angle a approaches 90 degrees, to needle-like, as it approaches 0 degrees. For angles near 90 degrees, we observe an isotropic fluid, a rhombatic fluid, a rotator phase, and a columnar space-filling structure with increasing density. Conversely, as a approaches 0 degrees, the results resemble the needle limit. Even for angles as small as a = 20 degrees, we still obtain isotropic, nematic, and rhombatic fluids before reaching a rhombic solid, but the nematic phase gains importance with decreasing a. At a approximately 60 degrees, aperiodic space-filling structures with long-range six-fold orientational symmetry dominate over periodic candidates such as the rhombic and rhombille. This aperiodic solid undergoes a melting process leading to a phase with quasi-long-range six-fold orientational symmetry, a hexatic fluid, before reaching the isotropic phase.

Hybrid superconductor-semiconductor systems with large spin-orbit coupling are important platforms for realizing topological or triplet superconductivity. Planar Josephson junctions made using these materials are predicted to enter the topological state by tuning the phase difference between the two superconductors from 0 to $\pi$. The 0-$\pi$ transition can be driven by magnetic field through Zeeman splitting of subbands in the semiconductor. It is expected to manifest as a node, or a re-entrance, in the critical current. Here we present re-entrant switching currents from several InAs/Al planar Josephson junctions in high in-plane magnetic fields. We find that re-entrances in some devices conform with expected signatures for topological or 0-$\pi$ transitions. However, we show that the data can also be explained in terms of mode interference in the junction in the presence of disorder. We also present simulations of supercurrent interference under in-plane fields that can reproduce re-entrances due to corrugated weak link without invoking the Zeeman effect or topology.

Gas bubble transport through highly porous transport layers (PTLs) is a key process in electrochemical devices such as proton exchange membrane water electrolyzers, where bubbles generated at catalyst surfaces must migrate through complex porous networks. To understand this process, we focus on model systems, namely the motion of single, paired and multiple bubbles in capillaries and study these by combining analytical modeling, three-dimensional color-gradient lattice Boltzmann simulations, and X-ray radiography. For single bubbles, we derive an analytical expression for the critical Bond number separating passage from clogging and show that, in the low deformation regime, it accurately predicts this transition in circular capillaries. Extending the study to bubble pairs, we uncover additional clogging and unclogging pathways, including hydrodynamic unclogging driven by pressure buildup in the interbubble film, and coalescence-induced clogging and unclogging. By mapping our results as functions of confinement ratio and Bond number, we define distinct dynamical regimes that control bubble passage. Experiments on bubble chains rising through highly porous nickel foams confirm the predicted clogging and unclogging mechanisms.

We propose concrete protocols to realize quantum criticality due to excited-state quantum phase transitions (ESQPTs) experimentally in presumably the simplest and most resilient system involving a single trapped ion oscillating in a radio-frequency Paul trap. We identify a specific class of excited states of the Extended Rabi Model (ERM) Hamiltonian, which occur between two critical ESQPT energies of the model in its (anti)Jaynes-Cummings superradiant phase. Properties of these states motivate the definition of several ESQPT witness observables. We study their critical scaling behaviors as well as various distinct state evolutions by driving the system across the quantum criticalities by changing the qubit-phonon coupling strength linearly in time at different finite rates. A mapping of the theoretical control parameters of the ERM to the experimental parameters of a trapped ion setup is provided, and simulations are performed for values referencing existing state-of-the-art setups, addressing both unitary state evolutions as well as relevant open-system corrections.

Flat bands have attracted considerable interest in condensed matter physics because they provide a fertile platform for realizing strongly correlated and topological quantum phases. To date, however, most studies have focused on flat bands in normal-state electronic structures, such as those found in graphene and transition metal dichalcogenides. In this work, we investigate the emergence of flat bands in the superconducting Bogoliubov quasiparticle spectrum of twisted layered $d$-wave superconductors. We show that when the superconducting order parameter is odd under the in-plane $\mathrm{C}_2$ rotation, Bogoliubov flat bands can be engineered in the vicinity of the rotation axis. By analyzing a low-energy effective Hamiltonian, we demonstrate that the Berry connection of single layer system provides a clear criterion for the formation of the Bogoliubov flat bands. Our results establish a new paradigm of superconducting twistronics, in which the twist angle acts as a powerful tuning parameter for designing gapless flat-band superconductors.

The performance of the variational quantum eigensolver depends critically on the choice of ansatz. In this work, we experimentally evaluate the emergent-coupling-based ansatz (ECBA), a physically motivated variational ansatz for disordered systems. The ECBA is based on a renormalization (semi-)group approach to determine the dominant effective couplings, resulting in shallow circuits that capture the essential long-range entanglement structure while balancing local correlations. We implement the ECBA on superconducting quantum processors and benchmark it on disordered Heisenberg chain models. Using classically pre-optimized parameters and error mitigation techniques, we study systems of up to 30 qubits and observe an experimental relative energy accuracy of 96.41% for the largest system. Furthermore, we find that the ECBA can be efficiently embedded on hardware with two-dimensional square-lattice connectivity. We compare to commonly used hardware efficient ansätze and observe that the ECBA achieves significantly higher accuracy at a similar gate count.

The escalating demand for memory scaling requires switching mechanisms that remain reliable at atomic thickness while operating with minimal energy consumption. Sliding ferroelectricity provides a promising platform for this challenge: the spontaneous interfacial polarization emerging at superlubric, atomically thin van der Waals interfaces endows exceptional fatigue resistance, ultrafast switching and ultralow coercive fields. Nevertheless, the intrinsically weak polarization of sliding ferroelectrics limits the available signal window, necessitating new physical mechanisms that can transduce subtle polarization variations into pronounced resistance contrasts. Here, we address this challenge by introducing momentum-conserving resonant tunneling between lattice-aligned graphene electrodes. The resulting resonant sliding ferroelectric tunnel junction achieves a tunneling electroresistance (TER) ratio of up to 225.65%, substantially exceeding that of conventional sliding ferroelectric tunnel junctions. In addition, the device delivers a tunable TER ratio, multistate programmability, high current density, robust endurance with a small coefficient of variation (<0.69%), fast switching (20 ns), low switching energy (310 fJ), and low read voltage (<0.2 V). Collectively, these results establish a unique role for sliding ferroelectricity in bridging the gap of memory technology between performance and miniaturization, and open a new pathway toward next-generation nonvolatile memory technologies.

Randomly-assembled dynamical systems are theoretically predicted to be unstable upon crossing a critical threshold of complexity, as first shown by May. Yet, empirical complex systems exhibit remarkable stability, indicating the presence of additional mechanisms playing a stabilizing role. The relation between complexity and stability is typically assessed by assuming fixed interactions, whereas real systems often evolve in intrinsically time-dependent states. To understand how this affects stability, we linearize a general non-autonomous dynamics around a reference operating state and model the resulting parameters as stochastic processes, which represent the minimal extension of static random interactions to time-varying ones. We derive exact stability bounds that generalize complexity-stability theory to dynamically varying systems. Notably, we find that temporal variability allows systems to remain stable even when their instantaneous Jacobian would predict instability. We compare our results against a non-linear neural network model, where our theory applies exactly, and the generalized Lotka-Volterra equations, where we numerically find that time-varying interactions systematically postpone the onset of replica-symmetry breaking. Overall, our results indicate that temporal variability systematically improves stability, demonstrating a general mechanism by which complex systems can violate classical complexity-stability bounds.

Molybdenum disulfide (MoS2) is a promising 2D transition metal dichalcogenide (TMD) for optoelectronics and quantum technology applications, but faces challenges in scalable synthesis and defect engineering. Oxygen-assisted chemical vapor deposition (O-CVD), which introduces in-situ oxygen during growth, shows excellent potential in resolving both issues at once. Although co-flowing oxygen shows improvement in growth, the underlying mechanistic role of oxygen remains unclear. In this work, a combination of oxygen dosing experiments, density functional theory (DFT) calculations, computational fluid dynamics (CFD) simulations, and ab initio molecular dynamics (AIMD) simulations, uncover the dual role of oxygen in O-CVD. Firstly, AIMD reveals that oxygen increases MoO3 sublimation and enhances Mo3O9 supply. Concomitantly, DFT reveals that sulphur oxides, due to their bulkier nature than pure S2, limit the formation of reactive MoS6 intermediates. Subsequently, by experimentally varying the oxygen flow-interval, flow-rate, and flow-time, and correlating them with CFD simulations, we decouple oxygen's roles in source-poisoning prevention (i.e. MoO3 evaporation) and growth regulation. We find that maintaining a low sulphur-to-oxygen (S:O2) ratio at the MoO3 boat and substrate during nucleation, and a high S:O2 ratio at the substrate during growth is the key to obtaining large-area high-quality monolayer MoS2, confirmed by our optical measurements. Based on our understanding, we present a kinetic phase diagram for MoS2 synthesis, which will enable controlled oxygen dosing as a tuning parameter for scalable, defect-controlled monolayer MoS2 synthesis.

The conjectured three generic local bulk statistics amongst all non-Hermitian random matrix symmetry classes have recently been extended to three generic local edge statistics. We study analytically and numerically complex spacing ratios and nearest-neighbour (NN) spacing distributions that characterise such local statistics. We choose the three simplest representatives of these universality classes, given by the Gaussian ensembles of complex Ginibre, complex symmetric and complex self-dual matrices, denoted by class A, AI$^\dag$ and AII$^\dag$. In the first part, we analytically study the complex spacing ratio in class A, at finite matrix size $N$. Introducing a conditional point process, we simplify existing expressions and show why an uncontrolled approximation introduced earlier converges well in the large-$N$ limit in the bulk. When specifying to the elliptic Ginibre ensemble, we present a parameter-dependent $N=3$ surmise for the complex spacing ratio, interpolating to that of the Gaussian unitary ensemble (GUE), where such a surmise is very accurate. In the second numerical part, we compare complex spacing ratios, its moments, and NN spacing distributions for all three ensembles with that of uncorrelated points, the two-dimensional (2D) Poisson process, both in the bulk and at the edge. The varying degree of repulsion within these different edge universality classes can be well understood in terms of an effective 2D Coulomb gas description, at different values of inverse temperature $\beta$. We find indications that the complex spacing ratio does not fully unfold the local statistics at the edge. Finally we verify that for small argument, in all three symmetry classes the NN spacing distributions in the bulk and at the edge are consistent with the universal cubic repulsion.

Optical cavities enable strong, long-range, light-matter interactions that can drive collective ordering phenomena, such as superradiant self-organization in ultracold atomic gases. Extending these ideas to solid-state electron systems could enable continuous-wave optical control of electronic order, but is impeded by the mismatch between optical wavelengths and electronic length scales. Here, we propose a platform for realizing superradiant charge density waves (sCDWs) in doped, driven transition-metal dichalcogenides coupled to an optical cavity. A nanoscale grating generates electric fields at large in-plane optical momenta, allowing cavity photons to couple efficiently to electronic density fluctuations through exciton-polaron processes. Using a linear-stability analysis, we determine the threshold for superradiant ordering and map out the driven phase diagram. We show that tuning the grating periodicity to match the enhanced electronic density fluctuations - such as those near Wigner crystallization - substantially lowers the required pump intensity. Our results establish a novel route toward cavity-controlled electronic order in quantum materials.

The exchange bias (EB) in ferromagnetic/antiferromagnetic (FM/AFM) bilayer systems causes a shift of the magnetic hysteresis curve after field cooling through the Néel temperature of the AFM. In some cases, this shift is accompanied by an asymmetry between ascending and descending branches. In the past, this asymmetric magnetization reversal has been studied in different bilayer systems, including polycrystalline Co/CoO thin films. Here we investigate the asymmetric magnetization reversal in an epitaxial fcc-Co(001)/CoO(001) thin film grown on MgO(001) by molecular beam epitaxy at varying temperatures up to room temperature after field cooling along the easy and the hard axes. Room temperature measurements of the longitudinal and transverse magneto-optic Kerr effect show different magnetization reversal processes via stable intermediate states for different angles between external magnetic field and magnetic easy axes. Once the sample is cooled below the blocking temperature, a pronounced asymmetric magnetization reversal can be observed. We show that in contrast to polycrystalline bilayers, the loop asymmetry stays constant after multiple training cycles and that the magnitude of the asymmetry is directly correlated with the magnitude of the EB.

Microscopic insights into nonlinear interactions are essential for advancing polaritonic devices. Existing studies often rely on phenomenological models that overlook important many-body processes. Based on a material-specific and predictive approach, we investigate monolayer and homobilayer MoS$_2$ embedded in a Fabry-Pérot cavity to characterize the exchange, saturation, and dipole-dipole contributions to polariton-polariton interactions in these technologically promising materials. A key finding is that the exchange interaction induces asymmetric energy shifts of the lower and upper polariton branches in a detuned cavity, a behavior driven by the difference in their excitonic character. Furthermore, we demonstrate that temperature and electron-photon coupling determine the energy renormalization through the equilibrium polariton distribution. In homobilayers, the dipole-dipole interaction is mediated by the interlayer character, enabling electrical control and facilitating the electric-field-induced closing of anti-crossings due to dipolar-interaction shifts. The gained insights on polariton-polariton interactions are important for the development of ultra-compact polaritonic circuitry.

Fivefold-twinned metal nanoparticles host a central wedge disclination that strongly influences their mechanical and catalytic properties. Yet the atomistic mechanisms governing the stability, migration, and annihilation of this topological defect remain incompletely understood. Here we present a systematic molecular dynamics study of gold Marks decahedra in which the fivefold axis is artificially brought close to the surface by controlled geometric modifications. By generating concave and convex morphologies with varying axis depth, we uncover a geometry-controlled competition between axis centering and detwinning. Concave geometries promote surface diffusion that restores fivefold symmetry, either by recentering the original disclination or by nucleating a new subsurface axis through collective atomic rearrangements. In contrast, convex structures with a shallow axis undergo rapid detwinning within the first nanoseconds via surface glide, leading to single-twin or fully FCC configurations. Remarkably, positioning the axis just two atomic layers beneath the surface suppresses detwinning and restores stability. Our results demonstrate that surface curvature and defect depth critically regulate disclination mobility and twin stability, providing a mechanistic framework to understand the structural evolution of multi-twinned nanoparticles and to guide the controlled design of defect-engineered nanomaterials.

Disordered fiber networks exhibit a floppy to rigid mechanical phase transition as a function of connectivity. Sub-isostatically connected networks can undergo this transition via straining. Critical exponents governing this transition have been estimated theoretically and by numerical simulations of various types of networks. In this study, we present improved results, achieved through a combination of refined numerical simulations, larger system sizes and incorporation of theoretical predictions for better post-simulation analysis. We also report the evolution of the critical strain and critical exponents as the network is sheared while being subjected to non-volume-preserving uniaxial deformations.

The explicit expression of ergotropy (a.k.a. available energy) of a classical system is known for the case when the system phase space density is continuous and with no plateaus. Here we provide the general expression of ergotropy that applies without those limitations. It easily follows upon casting the ergotropy problem as a function rearrangement problem. This leads to the notion of "ergotropic rearangement" which generalises that of "symmetric decreasing rearrangement" (an advanced topic of measure theory). We apply it to investigate the fate of classical ergotropy in the thermodynamic limit, and find that any density of the form $\rho=f(H_0)$ is asymptotically passive, where $H_0$ is the system Hamiltonian and $f$ a generic function.

Artificial spin ice provides a versatile setting for emergent gauge fields and magnetic monopole excitations. Here we propose a driven-dissipative polariton realization of artificial spin ice, in which the circular polarization of each link mode plays the role of an Ising degree of freedom, while an auxiliary lossy vertex mode dynamically enforces a local ice-rule constraint. Adiabatic elimination of the vertex mode yields an effective spin-ice penalty, favoring the two-in two-out manifold in the steady state. We show that local polarization flips generate monopole-antimonopole defects, and that sequential flips transport these defects across the lattice while defining a Dirac string. In an extended spin-ice geometry, the vertex charges and their dynamics can be directly reconstructed from polarization-resolved real-space imaging. Our results establish polariton lattices as a controllable photonic platform for creating, manipulating, and observing emergent gauge charges in nonequilibrium spin-ice systems.

Simulating cavity quantum electrodynamics in synthetic platforms offers a route to studying light–matter physics without real photons. We reassess the frequently invoked analogy between harmonically trapped, spin–orbit-coupled atoms and the quantum Rabi and Dicke models, with particular emphasis on squeezing and entanglement. A single non-interacting atom realizes the quantum Rabi Hamiltonian exactly, up to elementary spin and phase-space rotations. For $N$ non-interacting atoms, however, the center-of-mass mode is only one member of a complete orthonormal set containing $N-1$ relative modes. We prove that, after all modes are included, the spin–spin terms generated by the collective mode cancel exactly against those generated by the relative modes. Equivalently, the original Hamiltonian is a sum of independent one-particle Hamiltonians and therefore cannot create interparticle entanglement from a separable state. Contact interactions invalidate this cancellation but do not remove the relative modes or turn the exact many-body Hamiltonian into a Dicke model. A one-mode Dicke description can arise only as a controlled low-energy approximation after all relevant relative excitations have been shown to be spectrally suppressed. We discuss this distinction together with experimental scales, finite temperature, and possible mode-selection strategies. We also distinguish the single-particle interference underlying density fringes from the interaction-stabilized stripe supersolid observed in many-body condensates.

Laser-induced magnetization precession measurements in ferromagnets often reveal an anomalous decrease in the damping time near a field-induced second-order spin-orientation transition, a behavior that cannot be described by the linearized Landau-Lifshitz-Gilbert equation. Here we demonstrate that this anomaly is not a material property but results from interference of precessing local magnetizations within the inhomogeneously excited region. By combining pump-probe experiments, analytical modeling that accounts for the finite sizes of the pump and probe spots, and micromagnetic simulations, we show that the standard macrospin approach fails to capture the observed dynamics. The inhomogeneous relaxation of magnetic parameters within the excitation area distorts the measured precession envelope, while dipole fields give rise to a temporally non-monotonic term in its frequency. Our results highlight the critical role of excitation locality in a vicinity of critical fields.

Multimodal locomotion is crucial for an animal's adaptability in unstructured wild environments. Similarly, in the human gastrointestinal tract, characterized by viscoelastic mucus, complex rugae, and narrow sphincters like the cardia, multimodal locomotion is also essential for a small-scale soft robot to conduct tasks. Here, we introduce a small-scale compact, foldable, and robust soft electromagnetic robot (M-SEMR) with more than nine locomotion modes designed for such a scenario. Featuring a six-spoke elastomer body embedded with liquid metal channels and driven by Laplace forces under a static magnetic field, the M-SEMR is capable of rapid transitions (< 0.35 s) among different locomotion modes. It achieves exceptional agility, including high-speed rolling (818 mm/s, 26 BL/s), omnidirectional crawling, jumping, and swimming. Notably, the robot can fold to reduce its volume by 79%, enabling it to traverse confined spaces. We further validate its navigation capabilities on complex terrains, including discrete obstacles, viscoelastic gelatin surfaces, viscous fluids, and simulated biological tissues. This system offers a versatile strategy for developing high-mobility soft robots for future biomedical applications.

We study the instability in RuO$_2$ using the Hartree-Fock approximation followed by the random phase approximation. We employ a three-orbital Hubbard model without spin-orbit coupling. An analysis of the eigenvalues and eigenvectors of the static susceptibility in the non-magnetic phase for various local interaction parameters $U$, $J_H$, and hole doping $n$ shows that the spin susceptibility is the dominant response channel. In the stoichiometric system without spin-orbit coupling, commensurate altermagnetic order is identified as the leading instability at sufficiently low temperatures, whereas at higher temperatures or finite hole doping, incommensurate wave vectors emerge. To elucidate the origin of the magnetic instability, we analyze the band spitting by the staggered Weiss field and discuss the qualitative difference between altermagnets and antiferromagnets.

Chemical waves of CO oxidation on platinum surfaces exhibit complex spatio-temporal self-oscillations, yet the local electronic mechanisms driving their propagation remain poorly understood under operando conditions. In this work, we combine operando scanning electron microscopy with frequency-modulated Kelvin probe force microscopy (FM-KPFM) to simultaneously map secondary electron contrast and local work-function variations during CO oxidation on Pt. By utilizing the KPFM tip as a localized sensor, we provide the first work-function-resolved imaging of reaction fronts, enabling an unambiguous physical assignment of CO- and oxygen-covered states. Our results demonstrate that the spillover process of chemical wave-the transition and expansion of adsorbate phases-is characterized by a pronounced temporal asymmetry and spatial heterogeneity transition thresholds. KPFM identifies a rapid onset of oxygen coverage followed by a gradual, diffuse relaxation back to the CO-covered state, indicative of relaxation-type oscillations even at low pressures (10^-2 Pa). Correlative reaction-diffusion simulations reproduce this wave morphology, confirming that the high-resolution work-function signal provides unique insights into the internal structure and kinetic heterogeneity of the working catalyst surface.

We study the conditions under which Matrix Product States (MPS) or Matrix Product Operators are exact eigenvectors of an extensive local operator, such as a Hamiltonian. By suitably choosing the local operator, this covers a wide range of settings: Exact eigenstates of Hamiltonians, including scar states, exact MPS trajectories for driven quantum systems, steady states of local Lindbladians, generalized symmetries of either Hamiltonians or density matrices, and many more. Our key result is that a local, fixed-size equation—namely, how a single term in the operator acts on a block of tensors—provides a necessary and sufficient condition for exact solutions. This local characterization allows us to obtain the full space of MPS solutions to all of the aforementioned problems; in particular, we exemplify the power of our results by recovering the quantum group symmetries of the XXZ model. We also discuss applications to numerical algorithms with MPS and the generalization of our results to 2D, i.e., projected entangled pair states (PEPS).

We develop a geometric formulation of stochastic dynamics in which noise, diffusion, path probabilities, fluctuation theorems, and entropy production arise from the intrinsic geometry of an evolving manifold rather than from externally imposed randomness. Within the theory of moving manifolds, we establish a curvature-noise correspondence: fluctuations are governed by the inverse curvature tensor, while entropy production is controlled by curvature deformation. The invariant continuity law on a moving hypersurface yields a geometric Fokker-Planck equation, and curvature-velocity coupling generates a quadratic Onsager-Machlup functional determining path weights. The resulting entropy functional satisfies a curvature-driven monotonicity law, providing a geometric derivation of the Second Law. In two dimensions, the curvature invariant reduces to Gaussian curvature and encodes topology, so topological transitions produce discrete entropy jumps. When the ambient space carries a Minkowskian signature, the same curvature-kinetic quadratic form that generates dissipative thermal weights produces oscillatory phase weights, and the Laplace-Beltrami operator governing entropy evolution acquires a Schrödinger-type structure. This provides a geometric resolution of the apparent distinction between classical stochastic behaviour and quantum dynamics. These results show that stochastic behaviour, thermodynamic irreversibility, and quantum transition amplitudes are unified within the moving manifold framework. Geometry does not merely accommodate stochasticity; stochastic behaviour arises as a consequence of deterministic geometric evolution. The theory predicts curvature-controlled anisotropic diffusion, entropy jumps at topology-changing events, and a geometric thermal-quantum crossover in which classical stochastic weights and quantum amplitudes are generated by the same curvature-kinetic action.