Papers

In the machine learning community, the reinforcement-learning-driven reasoning process has been intensively studied since around 2025. In this article, we present a negative result on the hardware implementation of this reinforcement-learning-driven reasoning process in a physical neural network model in the presence of a reward external potential energy for a given reward. In this hardware implementation, the existence of a chain of thought in dynamics contradicts the Lyapunov stability of the system.

Relevant IceCube recent data for astrophysical neutrinos are analyzed. Similarly to cosmic rays and proton-proton LHC collisions, the corresponding spectra are satisfactorily described, for the entire interval of measured energies, through q-Gaussians. This fact reinforces the already available evidence that nonadditive entropies (either of the $S_q$ class or of the $S_{delta}$ one, or even alternative ones) appear to play a crucial role in systems such as black holes and related cosmological phenomena (e.g., dark energy). The IceCube data yield sensibly high values for the momenta-distribution nonadditive index (q \simeq 1.6), in contrast with the cosmic rays values (q \simeq 1.2) and the LHC values (q \simeq 1.1). This fact neatly indicates very-long-ranged space-time correlations. The detailed associated physical mechanisms remain, at the present stage, as an instigating open issue.

AI-assisted scientific discovery increasingly seeks to move beyond generating plausible ideas toward reproducible, constraint-aware workflows that can support practical engineering decisions. We present MARS, a hierarchical Multi-Agent Reasoning System for knowledge-grounded material substitution under scientific and manufacturing constraints. We apply MARS to replace per- and polyfluoroalkyl substances (PFAS), a demanding materials-design problem in which substitutes must satisfy functional performance, environmental, experimental, and processing requirements. MARS decomposes substitution into a closed-loop workflow comprising requirement analysis, grounded candidate generation, and manufacturability assessment with process design. The system integrates retrieval-augmented generation, knowledge-graph reasoning, a database of lab-available materials, persistent memory of intermediate results, and rejection-aware iterative feedback. In a PFAS-substitution case study, MARS generates structured material requirements, identifies an evidence-grounded substitute candidate, and develops a lab-scale manufacturing route. In a multi-seed ablation evaluation, MARS achieves the highest overall performance across all three subsystems, with gains that persist when the backbone model is held fixed. These results suggest that hierarchical decomposition, knowledge grounding, and manufacturability-aware refinement improve material-substitution reasoning beyond model capability alone.

Scattering of quasiparticles provides direct access to the interactions that govern strongly correlated systems. In this work, we realize the quasiparticle scattering experiment of the Fermi-Hubbard model on a digital quantum processor by colliding a doublon and a holon in the half-filled chain of 21 sites encoded on 42 qubits of the ibm_boston processor. For the first time, we measure the transmission probability of quasiparticles of the Hubbard model on quantum hardware, which agrees with the Bethe-ansatz scattering matrix without free parameters. Beyond the scattering probabilities, the collision generates entanglement by superposing the transmitted and reflected components, which we infer from the particle-number fluctuation. In sharp contrast, a control collision between two doublons shows neither the interaction dependence nor the number fluctuation, as expected for indistinguishable particles. Our work demonstrates that the ability to create quasiparticles of a chosen flavor on demand and program their motion and interactions makes a digital quantum processor a new experimental toolset for probing the entanglement structure generated by quasiparticle scattering.

Quantum mechanics seems to be an incredibly delicate theory. Slight alterations to nearly any part of its structure can lead to huge qualitative differences and wildly unphysical predictions. Here, we explore the many-body consequences of replacing the Born rule $p(m)\propto\|P_m|\psi \rangle|^2$ by a warped rule $p_b(m)\propto\|P_m|\psi\rangle\|^{2b}$. Specifically, in systems with conserved particle number, evolution punctuated by repeated local particle observations acquires a probabilistic bias over measurement histories with an effect controlled by the sign of $b-1$. For $b>1$, this bias generically drives the gas to collect into nearly frozen configurations. Similar to the action of Maxwell's demon, the simple act of observation seems to extract entropy from a closed system. For $b<1$, the bias is reversed, generically scouring out long-wavelength fluctuations to produce a hyperuniform gas. We study this effect in a one-dimensional quantum gas, where these two phases are derived analytically for a particular measurement scheme and corroborated numerically for more general procedures with discrete-time projective-measurement circuit models using matrix product state simulations. We additionally discuss how this bias manifests in a common technique that often employs this warped rule, the “replica trick,” implying qualitative differences between finite-replica and replica-limit calculations.

The $q$-deformed Weil-Petersson volumes of Do and Norbury are the top-degree parts of Okuyama's discrete volumes, which are defined by topological recursion on the spectral curve of the double-scaled SYK (DSSYK) model. We show that the same q-deformed volumes can be obtained from a double-scaled matrix model whose disk density is found by zooming in on the soft edges of the DSSYK density. Applying the same procedure to the hard edge of $\mathcal{N}=1$ DSSYK, we obtain candidate $q$-deformed $\mathcal{N}=1$ super Weil-Petersson volumes. They are polynomials of fixed degree, and they reduce to the $\mathcal{N}=1$ super Weil-Petersson volumes in a scaled $q\to1$ limit. These candidate volumes differ from the $\mathcal{N}=1$ volumes of Do and Norbury in (arXiv:2510.12431) up to what we call "irrelevant" contributions which exactly vanish in the scaled $q\to 1$ limit. Together, these results demonstrate how q-deformed volumes can be naturally associated to the spectral edges of various DSSYK models and provides an intuitive way to construct them in supersymmetric contexts.

Current AI predicts materials by speeding up energy minimisation and training on large databases. However, scaling data generation prevents manual analysis by domain experts. If novelty is evaluated by chemical compositions or traditional representations in Crystallographic Information Files, or standard structure matchers based on manually chosen thresholds, one can easily generate millions of “new” compositions or files without discovering truly novel materials. This paper uses a hierarchy of crystal invariants to continuously trace and quantify structural novelty based on the new mathematical definition of a crystal structure and asymptotic guarantees for invariant descriptors. The invariant-based analysis through cascade computations on a modest computer exposes the unexpected scale of duplication in the world's largest materials databases, shown here in the high-profile case of GNoME datasets. To prevent current “embarrassment of over-prediction”, we propose necessary conditions for next-generation AI that should be automatically enforced before claiming any new materials.

We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-$\sigma$ KdV charge inserted with chemical potential $\alpha_\sigma$. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the $c=1/2$ free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-$5$ deformation for the first few charges evaluated on low-lying states.

We investigate the rate of change of equilibrium potential energy for nonlinear elastic solids undergoing evolutions that may involve changes in body geometry, material properties, applied tractions, and prescribed displacements. We represent such an evolution by a one-parameter family of variational problems and identify a class of these families for which the potential-energy release rate is well defined. Particular attention is given to the formulation of the potential energy at each evolution state: each constituent variational problem must yield the appropriate equilibrium state, while its equilibrium value must also correctly represent the physical change in potential energy between states. Within this framework, we derive a closed-form expression for the potential-energy release rate for solids with bounded domains, allowing for spatially nonuniform material evolution, sharp material interfaces, and evolving boundary geometry and prescribed boundary data. The expression is determined entirely by the equilibrium solution at the state under consideration and by the quantities prescribing the local evolution, so neighboring equilibrium problems need not be solved. Finally, we apply the result to the extension of a notch in a two-dimensional elastic body and show that the resulting potential-energy release rate reduces to the classical $J$-integral.

New physics emerges from new materials, which in turn rely on the development of new synthesis techniques. Identifying the enlargement of metastable synthesis window as enabling for realizing novel materials, we develop diffusion-activated epitaxy, a technique which combines two metastable synthesis methods of diffusion and epitaxy. We observe that within a narrow window of investigated synthesis space we can experimentally identify the stabilization of four distinct phases, three of which show X-ray diffraction and scanning transmission electron microscopy characterization data that are incompatible with the crystal structures registered in experimental and computational inorganic structure databases. These results suggest an enlarged metastable synthesis window is enabled by this synthesis approach, identifying doubly metastable synthesis as a promising method for exploration of quantum materials.

We find a robust scenario for the emergence of the Golden Ratio (GR) dynamical universality class, characterized by the dynamical exponent $z \equiv \varphi=\frac{1}{2}(1+\sqrt{5})$. Previously, its emergence required fine-tuning of system parameters. Here, the GR universality class appears generically in a wide class of Markov processes as a consequence of a global, circulant symmetry of the flux Jacobian. This symmetry leads to GR universality in systems where the total number of conservation laws is odd and larger than three. We introduce a multilane exclusion process with Bernoulli stationary measure as a minimal, parameter-free model, which features GR universality for all conserved modes, most of which are doubly degenerate. Our predictions, based on mode-coupling theory generalized to treat degenerate conserved modes, are corroborated by large-scale Monte Carlo simulations.

Phononic frequency combs are equidistant spectral ladders generated when a driven vibrational mode transfers energy to an autoparametrically excited subharmonic mode through anharmonic intermode coupling. In previous studies, the coupling strength governing this energy transfer has typically been treated as a free parameter, extracted from a fabricated device, or adopted from an earlier calculation for a specific mode pair. As a result, whether a given crystal can support a phononic frequency comb, and the drive strength required to generate it, has not been predictable a priori. Here we obtain the complete parameter set from density functional theory. Twisted bilayers of \MoSSe{} and \MoSMoSe{} at the commensurate angle $21.787^{\circ}$ adopt space group $P3$ in a 42-atom \rootseven{} moiré supercell. The second-order force constants provide the mode frequencies; the third-order constants provide the linewidths through the Boltzmann transport equation and hence the quality factors; and the same third-order tensor projected onto the phonon eigenvectors of a chosen mode pair provides the anharmonic couplings $\kt$ and $\kss$. Both bilayers constitute a mode pair within $0.8\%$ of the $2\!:\!1$ resonance with $\omega_1/\omega_2 = 1.9923$ in \MoSSe{} and $1.9943$ in \MoSMoSe{}. By integrating the resulting two-mode equations, both modes generate combs through a discontinuous transition: at $701\,\fth$ in \MoSSe{} the envelope variance rises from $2.1\times10^{3}$ to $1.37\times10^{4}$ and the tooth count of mode 1 from 7 to 16 within a single $5\,\fth$ increment, identifying the onset as subcritical. The spectral entropy peaks before this jump, at $341\,\fth$. The two compounds carry comparable coupling, differing by a factor of $1.2$. However, the comb excitation threshold is set by the quality-factors of signal modes: $Q_2 = 672$ for \MoSSe{} and $Q_2 = 3844$ for \MoSMoSe{}.

Performing measurements on gapless quantum ground states can give rise to new universal phenomena, whose origins remain poorly understood even for the paradigmatic Ising critical point. Replica methods offer a natural framework to study such post-measurement ensembles, with the replica limit $Q\to 1$ describing physical Born-averaged observables. Here we show that, for the monitored critical Ising chain, the replicated post-measurement ensemble of states itself undergoes a phase transition precisely at the physically relevant replica limit $Q \to 1$, preventing standard replica treatments. We use boundary conformal field theory (BCFT) methods to explain the underlying mechanism of this transition. Our methods yield a perturbative analytic calculation of the universal $O(1)$ constant in the Shannon entropy of the measurement record, previously known only numerically. We also apply our tools to study measurement-averaged spin correlators, providing a unified description of the universal properties of measurement outcomes and post-measurement states in the critical Ising chain.

Dopants can tune the activity of zirconia-based CO$_2$ hydrogenation catalysts through changes in adsorbate and transition-state (TS) energetics. Computing these quantities with density functional theory (DFT) becomes expensive when several dopants, reaction intermediates, and TSs must be screened. Here, we further develop the graph-based machine learning method Wasserstein Weisfeiler-Lehman Gaussian Process Regression (WWL-GPR): we introduce additional electronic-structure and local-environment features relevant for oxide surfaces and use the resulting model to predict adsorption and TS energies on ZrO$_2$(101) surfaces doped with Al$^{3+}$, Cd$^{2+}$, Ga$^{3+}$, In$^{3+}$, and Zn$^{2+}$. Adsorbate-surface structures are represented as attributed graphs that combine bonds connectivity and features. We benchmark WWL-GPR against thermodynamic scaling relations (TSR) for reaction intermediates and Bronsted-Evans-Polanyi relations (BEP) for TSs, and then propagate the predicted energies to microkinetic CH$_3$OH-yield estimates. WWL-GPR consistently improves the prediction of reaction intermediate energetics relative to linear scaling approaches and provides TS estimates that are competitive with BEP-based models. These improvements become especially relevant when the predicted energetics are propagated through the reaction network. Microkinetic calculations show that WWL-GPR-derived energies give CH$_3$OH yields closer to DFT-derived yields than the linear-relation workflow. Degree-of-rate-control analysis links the activity errors to dopant-dependent rate-controlling steps, including HCOO* formation, H$_2$COOH* dissociation, and H$_3$CO* formation. These results support WWL-GPR as a data-efficient surrogate for connecting atomistic energetics with catalyst-screening observables on doped oxide surfaces.

Dissipative state preparation engineers quantum many-body states as steady states of an open-system dynamics. The spatial range of the required dissipation sets both the feasibility of analog implementations and the connectivity and control complexity of digital ones. Yet how this locality quantitatively constrains preparation times remains largely unknown. Here, we establish an exact locality-preparation time relation for quadratic fermionic systems and develop a construction that optimizes the locality of the dissipative dynamics while preserving the target state. Applied to the Kitaev chain, our framework reveals sharply different locality-preparation time tradeoffs in critical and gapped regimes. At criticality, suppressing the algebraic tails of the jump operators necessarily forces the preparation time to grow algebraically with system size. By contrast, both trivial and topological gapped phases admit exponentially localized dissipation with a finite preparation time in the thermodynamic limit. Our results identify when fast dissipative preparation is compatible with local control and when criticality imposes an unavoidable dynamical cost.

We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.

A novel physics-informed neural networks (PINNs) based approach for solving functional renormalization group flow equations is presented. We investigate the finite temperature phase transition in $O(N)$ models without relying on the large-$N$ approximation, which was used in earlier attempts with similar methods. We argue that these previous solutions cannot easily be generalized for more complex symmetries, and hence our approach can provide new insight. Our technique relies on a series of ansätze that combines the small- and large-field approximation schemes with a neural network, leading to a highly accurate approximation over the entire field range. We compare the PINNs approach with traditional grid-based Runge-Kutta results and show that the network can successfully learn the temperature, field, and scale dependencies of the effective potential in the local potential approximation.

Frustration generated by competing interactions can stabilise patterns with tunable lengthscales which evade indefinite coarsening and macroscopic phase separation. Here we investigate a charged binary fluid in which the tendency toward phase separation is opposed by screened electrostatic repulsion. Combining analytics and numerical simulations, we identify a regime of arrested phase separation characterised by finite-wavelength instability. The resulting states include droplet and labyrinthine phases in two dimensions, and droplets, concentric shells and tubular structures in three dimensions. Surprisingly, some of these structures are amorphous morphologies that emerge reproducibly from different initial conditions. Free-energy estimates suggest that these disordered structures may be thermodynamically stabilised by frustration, rather than being kinetically trapped. Our results identify a simple mechanism through which the competition between phase separation and screened electrostatics can generate equilibrium amorphous microphases with glassy features.

A plate withdrawn from a bath of partially wetting liquid entrains a film above a critical velocity. Here, we show that this critical threshold can be substantially increased by the presence of a laminar gas flow opposed to the plate velocity that exerts a tangential shear stress on the liquid-gas interface. Using two-phase immiscible Navier-Stokes simulations and confining the gas to a narrow channel, we derive a closed-form expression for the interfacial stress, which combined with lubrication theory relates the thickness of the film and the contact line velocity to the applied shear stress. As the gas velocity is increased, the film thins and the contact line velocity slows down, until the imposed shear overcomes the drag of the moving plate and entrainment is arrested. As in the quiescent case, the velocity mismatch between plate and contact line is independent of the plate velocity and is set by the applied shear alone. We find that one set of varying gas flow simulations, with constant critical plate velocity, therefore determines the critical gas flow for every smaller plate velocity.

We investigate the anomaly-induced transport properties of a holographic superconductor exhibiting spontaneous symmetry breaking. The holographic model incorporates vector and axial gauge fields, an additional non-anomalous $U(1)$ symmetry, and a charged scalar field that drives the phase transition. Working in the probe limit, we numerically construct the superconducting background under two alternative scalar field quantizations, corresponding to operators with scaling dimensions $\Delta=1$ and $\Delta=3$ (denoted as the $O_1$ and $O_2$ theories, respectively). Using Kubo formulae, we compute the chiral magnetic conductivities and examine their response to the scalar condensate. Our findings demonstrate that spontaneous symmetry breaking allows anomaly-induced transport to extend beyond anomalous currents, significantly impacting the non-anomalous sector as well. Remarkably, this effect appears in both theories, indicating that the influence on the non-anomalous sector is independent of the chosen quantization. The comparison of our numerical results with the analytical limits for $O_i \to 0$ and $O_i \to \infty$ $(i=1,2)$ reveals consistent behavior across both theories.

Grain boundaries (GBs), including twin boundaries (TBs), play a critical role in plastic deformation by interacting with dislocations. The deformation and strengthening mechanisms in $\beta$-Sn are of significant interest because Sn-based alloys are widely used as solder materials. Recent experiments have shown deformation-induced faceting in $\beta$-Sn arising from twin-dislocation interactions. Using molecular dynamics (MD) simulations informed by a recently developed machine-learned interatomic potential (MLIP), we demonstrate a $\langle c\rangle$ dislocation-absorption mediated faceting reaction along $(301)$ TBs in $\beta$-Sn in which facet length grows with the number of absorbed dislocations. We show that the predicted facet length is consistent with isotropic elasticity theory only when the disclination character of the asymmetric twin facet is taken into account. We discuss analogies to faceting reactions in other low symmetry metals and hypothesize that the ubiquity of $(301)$ twins in $\beta$-Sn stems from their ability to grow rapidly through a forest of dislocation obstacles.

We develop an analytical approach to the Kondo problem based on an effective partition function that generalizes the sophomore's dream integral. The Kondo crossover is represented as a passage through an approximately equidistant ladder of its Lee-Yang zeros in the complex plane, rather than as a singularity on the real axis. Summation over this ladder yields the Gamma function $\Gamma(\alpha + T_K/(2 \pi T))$ as the universal crossover backbone, where the offset $\alpha =1/2$ is fixed by the Friedel sum rule. The Euler-Maclaurin expansion connects the discrete zero structure to the Nozieres Fermi-liquid theory through Bernoulli numbers. A two-cutoff formula provides a compact, essentially one-parameter description of the full crossover in terms of a temperature-space Fano lineshape which is a thermal average of the energy-space Fano resonance observed in STM. Drift corrections are treated. The approach extends to $\mathrm{SU}(N)$ Coqblin-Schrieffer, underscreened and Anderson impurity models, and Kondo lattices.

Complex disordered conductors can display highly structured and parameter-sensitive transport even in the absence of a thermodynamically stable glass phase. Overlap distributions originally developed for replica symmetry breaking (RSB) provide a useful diagnostic of this organization, provided that their interpretation is distinguished from the equilibrium Parisi construction. In this study, we report RSB-like overlap phenomenology and parametric mesoscopic glassiness in a graphene-based electronic Lévy glass, a device that mimics optical phenomena via electronic processes. By tuning the Fermi energy, we can control the number of propagating wave modes and observe reentrant changes from narrowly centered to broad or bimodal overlap distributions. The phenomenology is observed in both graphene-based and conventional two-dimensional electron-gas electronic Lévy glasses, indicating its generality. Sample-resolved overlap distributions show that the effect is intrinsic to each disorder landscape, whereas the normalized conductance variance shows that the growth of the absolute variance is mainly caused by the increasing mean conductance. We therefore interpret the overlap as a measure of correlation among hierarchically organized transport pathways and the reentrant behavior as evidence of pinning, parameter-induced reorganizations, and mesoscopic chaos in coherent transport.

Qubits are powerful computational resources because each additional qubit doubles the size of the accessible state space. This expressive power comes at a cost: adding qubits also increases circuit size, constraint overhead, and exposure to hardware noise. These limitations are especially critical for near-term quantum computers, making the number of qubits required to formulate a problem a central practical concern. In this work, we present a method for efficiently mapping logistics and routing problems onto qubits through structure-aware encoding reduction. We demonstrate the method on a benchmark problem of latency-resilient routing under dual-link failures. For this benchmark, the reduced formulation substantially lowers both the qubit count and the compiled circuit complexity while preserving the original optimization task. By running the reduced model on the same trapped-ion quantum computer as the benchmark, we obtain a 100X improvement in the success probability. This finding shows how careful problem formulation can make current quantum hardware substantially more effective for practical optimization.

We analyze ferroelectric domain wall motion, shape, and internal structure arising from coupling between polarization and screening charge. Universal cases are organized by electrical boundary conditions, charge conservation and dynamics, and polarization free energy. We consider exposed surfaces, interfaces beneath a dielectric layer and top electrode, and interfaces with a blocking mixed ionic-electronic conductor. Screening occurs through local exchange, exchange with lateral transport, or strictly conserved redistribution. Analytical derivations and finite-depth calculations establish screening drag, velocity branches, and conditions for spatial instability. For an isolated wall joining differently compensated domains, strict surface conservation and zero far-field current exclude steady translation of the screening profile, while allowing transient displacement, finite-sample arrest, and charge-balanced structural changes. Conservation also constrains long-distance transport around a moving junction in an adjoining conductor. Large-field simulations in the top-electrode geometry reveal nonlinear corrugation, transfer toward longer wavelengths, and coupled polarization rotation. First-order polarization energetics allow two interphase fronts enclosing a paraelectric core; multiaxial polarization permits wall rotation and competing intermediate ferroelectric domains. A driven structural-screening reduction supports stable periodic behavior. Reduced-variable diagrams summarize motion and structural regimes, while BaTiO3 and BiFeO3 examples connect them to material thermodynamics. Small-signal analysis distinguishes displacement, polarization, and electrostatic response functions. These results provide a framework for wall dynamics and structural transitions under different driving and screening conditions.

The Kelvin-Helmholtz (KH) instability is a fundamental signature of the natural world, manifesting in phenomena ranging from atmospheric clouds to the iconic turbulence of Jupiter's Great Red Spot. While the linear dynamics of this instability is well-understood, the ultimate fate of the fluid — the final stationary state to which it relaxes — remains a formidable challenge. In this paper, we present a general non-equilibrium statistical mechanics framework that allows us to predict the asymptotic vorticity distribution of a sheared 2D Euler fluid. By constructing a self-consistent ansatz for the solution of the Vlasov equation, we determine the stationary stream function without explicitly solving the time-dependent Euler equations. We demonstrate that the relaxation leads to a population-inverted core, where high-energy states are fully occupied, surrounded by a tenuous halo of vortices in the lower-energy regions of configuration space. This result establishes the core-halo structure as a robust statistical consequence of the system's evolution, offering a universal framework for understanding the saturation of shear-induced turbulence in inviscid 2D fluids.

We discuss three simple connections between generative diffusion and optimal transport. First, the joint law of the data and of their Ornstein-Uhlenbeck noising is, at every time, the solution of an entropic optimal-transport problem, at a temperature set by the noise level. Backward diffusion can consequently be read as an annealing which gradually decreases the entropic regularization. Second, we study the speciation transition–how the modes of the data distribution emerge during generation–and compare how it happens in diffusion and flow matching with respect to optimal transport. Using Gaussian mixtures as high-dimensional example, we show that the dynamical instability through which diffusion and flow matching separate the modes becomes, for optimal transport, an exponentially thin boundary layer at the beginning of generation, which is rounded by entropic regularization. Third, considering a generic forward dynamics with a general confining potential, we show that the backward process driven by the score also satisfies a drift-based variational principle whose minimal cost is the entropy difference between initial and final distributions.

We present, to our knowledge, the first theoretical realization of an independent charge-4e phase in a two-dimensional classical array of conventional superconductors coupled through altermagnetic junctions. Tuning the altermagnetic splitting suppresses the leading single-pair tunneling harmonic while retaining strong double-pair coupling, producing a finite charge-4e region between staggered and uniform charge-2e states. Using the complete temperature-dependent microscopic Josephson spectrum, we find short-range ordinary pair correlations alongside algebraic coherence of pairs of Cooper pairs. Berezinskii–Kosterlitz–Thouless scaling and a half-flux response further identify the charge-4e phase. Our results establish altermagnetism as a microscopic means to select between ordinary and composite superconducting coherence.

Transport networks are key to living and engineered systems. How can transport networks adapt efficiently using only local rules, without a global objective or specific boundary conditions? Inspired by observations in biological transport networks, we adapt networks by subsequently pruning edges by relative resistance, thus removing edges whose resistance is higher than the resistance for routing flow through the rest of the network. We find that adaptation by such a local, boundary-condition-free measure of redundancy, set by topology alone, yields sparse, loop-rich, low-dissipation networks.

Electrical transport in non-centrosymmetric materials exhibits a nonlinear conductivity tensor component that generates second-harmonic voltages in response to applied electric fields. Harnessing this effect is essential for developing zero-threshold electromagnetic energy harvesters and broadband frequency upconverters, both of which require maximizing the nonlinear responsivity. Here, we demonstrate that inhomogeneously strained macroscopic slabs of the topological insulator Bi$_2$Se$_3$ achieve a record-high nonlinear responsivity of $10^4\,\mu$m/V: a six orders of magnitude boost compared to its thin-film counterparts. We attribute this giant enhancement to a classical geometric nonlinear transport effect, which is driven by complex strain gradients within the bulk material. Notably, a similar geometric mechanism known as geometric nonlinear Hall was previously observed only in microscale-patterned structures. Our findings establish macroscopic strain engineering as a powerful route to enhance nonlinear transport, with immediate relevance to a broad class of layered systems, including transition metal dichalcogenides and moiré heterostructures.