Papers

Emergent systems - systems with multiscale interactions - arise through iteration than forward design, leaving principles for building them under-explored. Current artificial intelligence architectures, while useful for generation, provide no logic for construction. Inspired by statistical phylogenetics, we show that inferring scales of system entropy then constraining information from low to high entropic scales yields stepwise constructive models. We term this architecture the Layer-Restricted Boltzmann Machine (LRBM). Applied identically to digit images, natural language, and enzyme sequences, LRBMs follow a common hierarchical logic: lower layers encode global structure, higher layers fine-grained features. As a stringent test, we assayed 160 synthetic chorismate mutase (CM) constructive trajectories alongside 1,130 natural homologs in vivo. Designed CMs functioned up to 53% divergent from nearest natural homolog. Fold emerged at intermediate layers while function required all layers, illustrating that fold was necessary but insufficient for function. Our results demonstrate a shared, learnable logic for constructing emergent systems.

Neural Quantum States (NQS) provide a powerful neural network-based variational framework for representing many-body wave functions and solving for ground states. Recurrent Neural Networks (RNNs) are particularly promising owing to their relatively low computational cost and their autoregressive property, which enables perfect sampling. Recently, RNNs have been reported to be unstable under curvature-based optimizers such as the minimum-step stochastic reconfiguration (minSR) method. In this paper, we address this perceived limitation and show that minSR can be stabilized through simple regularization techniques, enabling robust training of RNN-based NQS with only a few samples. Our approach outperforms the Adam optimizer on the one-dimensional transverse-field Ising model and the one-dimensional cluster state, and provides competitive results on the two-dimensional Heisenberg and $J_1-J_2$ models. This work offers a promising pathway for using modern optimization techniques with autoregressive NQS to address open questions in quantum simulation.

Hypergraph (HG) and simplicial complex (SC) are two common representations of higher-order networks, and are often expected to differ mainly quantitatively when they encode comparable group interactions. Here we show that this expectation fails in evolutionary cooperation dynamics. Using a controlled higher-order public goods game (PGG) with minimal ad hoc parameters, we compare cooperation transitions on randomized HG and SC constructed from the same triangular backbone. We find that the impact of simplicial closure is selective: when the cooperation transition is continuous-like on HG, imposing simplicial closure mainly broadens the transition without changing its qualitative nature; however, when the transition is explosive and first-order-like on HG, simplicial closure will fragment the compact low/high bistability into a broad ensemble of metastable final states. Detailed analysis reveals that this selective effect arises from the dual role of simplicial closure in cooperative-nucleus dynamics: it promotes the survival of local cooperative nuclei while suppressing their conversion into system-wide cascades. Further experiments confirm that this fragmentation is not tied to a specific payoff form, but is a generic feature of explosive cooperation transitions in higher-order PGGs.

Anderson localization and the mobility edge (ME) have been extensively studied in isolated aperiodic systems. Conventional theory suggests that dephasing and decoherence should disrupt localization and facilitate transport. In this work, we investigate localization behaviors in a dimerized off-diagonal Aubry-Andre-Harper (AAH) quasicrystal subject to on-site pure dephasing. In the strong-dephasing limit, we apply adiabatic elimination within the Lindblad master equation framework to derive an effective classical Markov transition matrix that governs the dissipative relaxation dynamics. Counterintuitively, we demonstrate that pure dephasing can induce distinct MEs, including both conventional MEs separating extended and localized states and anomalous MEs separating multifractal critical states from localized states, even when all eigenstates of the original closed coherent system are delocalized or multifractal. Using fractal dimension finite-size scaling, wave-packet spreading dynamics, and energy spectrum statistics, we numerically verify the coexistence of fully extended, multifractal critical, and localized regions within the relaxation spectrum of the dissipative system, and construct the global dissipative phase diagram. These findings reveal that dephasing can see as a powerful mechanism for controlling localization transitions, thus enhancing our understanding of dissipative quasicrystal systems.

We study the multifractal criticality in a generalized Su-Schrieffer-Heeger model. The results show that the system supports not only critical phases but also re-entrance multifractalization (REM). By mapping the hopping term to an effective potential, we analytically prove that although the model has no explicit unbounded potential, a hidden unbounded potential is actually present-this is the key mechanism driving the emergence of multifractal critical phases. Moreover, one can get a condition where the competition between the explicit and hidden unbounded potentials is exactly balanced. Under this condition, the multifractal critical phase vanish, and the system returns to the extended phase. Based on this mechanism, we achieve both demultifractalization and re-entrant multifractalization. Finally, we double check the theoretical predictions through wave packet dynamics, and the numerical results are consistent with our theoretical analysis. This work broadens our understanding of how unbounded potentials induce multifractal critical phases, providing a theoretical basis for designing new systems with multifractal critical phases.

We formulate a non-Abelian theory of network compatibility in which dynamical transformations reside on the links. Gauge covariance follows from the freedom to choose local representation frames, while plaquette holonomies quantify the incompatibility of closed-loop transformations. For an SU(2) model on the complete simplicial 2-complex with quenched random plaquette couplings, parallel-tempering simulations reveal a continuous disorder-driven phase transition characterized by the network compatibility $M_P$. As the disorder strength approaches the critical value, the compatibility drops rapidly to a value that decreases with system size, while the frustration energy of the network sharply rises. Moreover, analysis of connected replica-overlap width provides no evidence for thermodynamic replica-symmetry breaking. Instead, the high-disorder regime sustains a nearly constant integrated adjacent correlation as the system size increases. Therefore, rather than a frozen gauge glass or a featureless disordered “gas” phase, dense topological frustration produces a non-glassy correlated gauge liquid in which individual pair correlations are geometrically diluted while a finite integrated correlation survives.

Hard carbons are established anodes for lithium-ion batteries and leading candidates for sodium-ion batteries, yet their electrochemical performance is governed by a heterogeneous network of graphitic domains, defects, and nanopores that conventional atomistic methods cannot model at the required length scales. We combine universal machine-learned interatomic potentials with the RAFFLE structure-generation framework to construct 13,096 realistic hard carbon models containing up to 4,378 atoms, matching experimentally measured densities, porosities, and sp$^2$/sp$^3$ bonding fractions. Explicit sodium intercalation of representative structures reproduces the characteristic sloping-to-plateau voltage profiles, revealing that capacity increases with decreasing carbon density and increasing porosity. To screen the full library, we train a lightweight neural-network surrogate that predicts capacity directly from the host using frozen universal-potential descriptors augmented by geometric void features. The surrogate identifies high-capacity candidates exceeding 800 mAh g$^{-1}$, which are validated by full intercalation calculations. This scalable framework links hard carbon microstructure to sodium-storage performance and provides atomistic design principles for high-capacity anodes. More broadly, the workflow enables systematic exploration of synthesis-dependent amorphous microstructures, including precursor chemistry, pyrolysis, heteroatom doping, and pore engineering, providing a route toward atomistically informed hard carbon design.

Singular kicked rotors have long been compared with power-law random banded matrices (PRBM) because their momentum-space Floquet matrix elements decay algebraically. However, it has remained unclear whether the deterministic correlations of the rotor become irrelevant at long distances and, consequently, under what conditions the two systems share the same infrared theory. To address this question, we derive a nonlocal supersymmetric nonlinear sigma model directly from a quantum kicked rotor with a power-law or logarithmic singularity. By carrying out the renormalization-group analysis up to two-loop order, we show that, after matching the symmetry class and coupling convention, the rotor reproduces the long-range Anderson transition of the corresponding PRBM, including its localized, critical, and extended infrared regimes.

Variational quantum sensing offers a promising route to high-precision parameter estimation, but its performance depends strongly on the circuit architectures used for probe preparation and measurement. Existing approaches typically optimize continuous parameters within predefined ansätze, restricting the accessible design space and limiting adaptation to sensing tasks and hardware constraints. Here, we introduce AutoQSense, a reinforcement-learning framework that searches circuit architectures using Fisher-information-based objectives. For few-qubit systems, a single agent sequentially constructs preparation and measurement circuits. For larger systems, a distributed formulation assigns local circuit design to subsystem agents and inter-block entanglement to a budgeted agent. Numerical results show that the learned architectures recover known benchmark strategies, adapt to dephasing noise, and outperform fixed hardware-efficient ansätze while using fewer entangling gates. These results establish AutoQSense as a resource-aware approach to adaptive and hardware-compatible quantum sensing.

The interplay between non-Hermiticity and localization has attracted considerable interest, yet the driven dynamics of localization transitions in non-Hermitian systems with on-site gain and loss remains largely unexplored. Here we investigate the critical scaling behavior and driven dynamics of the non-Hermitian Aubry-André (AA) model with on-site gain and loss. Through finite-size scaling analyses of the localization length, the inverse participation ratio (IPR), and the energy gap, we extract the critical exponents $\nu = 1.00(2)$, $s = 0.7965(2)$, and $z = 1.999(2)$. These exponents are different from those of both the Hermitian AA model and the nonreciprocal hopping AA model, particularly the IPR exponent $s$, demonstrating that the gain-loss mechanism belongs to a distinct universality class. For the driven dynamics, we focus on the case where the system is initially prepared in a gapless extended state and linearly driven across the critical point. We verify that the finite-time scaling (FTS) framework remains applicable provided that the criterion $z' < r$ is satisfied, where $z' = 1.999(2)$ characterizes the gap closure in the extended phase and $r = z + 1/\nu \approx 2.999$. The predicted FTS scaling forms for the IPR are numerically validated across a wide range of system sizes and driving rates, demonstrating that the unified scaling description can be successfully generalized to the gain-loss type non-Hermitian AA model. Our work not only establishes the gain-loss AA model as a new universality class of localization transitions but also extends the applicability of the FTS framework to non-Hermitian systems with gapless initial states.

Learning restricted Boltzmann machines (RBMs) is computationally challenging because it requires expectations whose exact evaluation is generally intractable. The expectations are typically evaluated using a sampling approximation based on blocked Gibbs sampling (BGS), which is a local Markov chain Monte Carlo transition kernel. However, the locality of BGS can lead to poor sampling quality when the RBM has high energy barriers, thereby degrading learning performance. Deep tempering (DT), which performs parallel tempering over a sequence of learnable RBMs including the training RBM, alleviates this locality issue. However, DT algorithmically requires multiple steps to move through the RBM sequence to achieve a nonlocal transition. In this paper, we propose a transition kernel defined over the RBM sequence used in DT. The proposed kernel has a round-trip structure over the sequence, enabling nonlocal moves within a single transition while leaving the RBM sequence invariant. Numerical experiments show that the proposed kernel performs nonlocal transitions more frequently and achieves higher sampling quality with fewer transitions than BGS and DT. We further verify that learning based on the proposed kernel is more stable and mitigates the training failures observed with BGS- and DT-based learning.

We explore kinetic-constraint induced thermodynamic phase transition in the cubic lattice, employing the Fredrikson-Anderson spin model with hyperparameter $K=2$ as a representative kinetic system. Each lattice site may flip its binary occupation state if at most one of its six nearest neighbors is currently occupied. The whole set of microscopic configurations that are kinetically connected with the fully empty one is described by an equilibrium partition function with a single global constraint, that is, the occupied sites do not form closed loops but instead organize into different tree components in the lattice. We discover a continuous thermodynamic gas–crystal phase transition in the cubic system and determine the critical chemical potential $\mu^* \approx -3.252$, at which the occupied sites of the equilibrium tree-packing configurations start to prefer one of the two nested cubic sublattices. This thermodynamic phase transition is absent in the two-dimensional square lattice.

We determine the distribution of fidelity susceptibility for chaotic eigenstates at the spectral edge of Gaussian random-matrix ensembles. Previous work showed that, in the unitary class, the characteristic susceptibility scale of edge states grows as $D^{1/3}$, rather than proportionally to $D$ as in the spectral bulk, reflecting Airy-edge level rigidity. Extending a determinant-based framework introduced for bulk states, we derive the universal edge distributions for both the orthogonal and unitary ensembles. The two symmetry classes share the scaling variable $g/D^{1/3}$ and exhibit a symmetry-dependent cubic suppression of small susceptibilities, while their algebraic large-$g$ tails reflect the corresponding symmetry-dependent level repulsion. Although eigenvector statistics retain their random-matrix form throughout the spectrum, edge rigidity makes low-lying chaotic states parametrically less sensitive to generic perturbations than bulk states. Our results establish universal, symmetry-dependent spectral-edge fidelity-susceptibility statistics in systems whose chaotic dynamics extends down to the ground state.

The mobility ring (MR), a critical boundary in the complex energy plane separating extended and localized states, is fundamental to understanding the Anderson transition in non-Hermitian (NH) disordered systems. While MRs have been extensively studied in one-dimensional (1D) NH quasiperiodic models, rigorous analytical frameworks beyond 1D remain critically scarce. Here, we investigate a class of two-dimensional (2D) quasiperiodically decorated Lieb lattices (QDLLs) featuring complex incommensurate potentials selectively applied to the lattice vertices. By exactly mapping these 2D structures onto NH generalized Aubry-André-Harper (AAH) models and leveraging extended-localized transition point, we analytically derive the Lyapunov exponents and obtain exact expressions for the MRs. These exact theoretical boundaries are strongly corroborated by numerical computations of wavefunction fractal dimensions and real-space probability distributions. Furthermore, we reveal distinct evolutionary behaviors of the MRs driven by the quasiperiodic potential strength: systems characterized by $\kappa=2$ possess a single MR, whereas systems with $\kappa=3$ undergo a dynamic sequential evolution from a single integrated ring into two independent rings. We hope that our exact results of MRs in 2D will benefit the study of Anderson localizations and MRs in high-dimensional NH systems.

A self-contained statistical-mechanics treatment of a single quantum harmonic oscillator is presented, whose frequency $\omega$ is drawn from a folded Gaussian distribution: $\omega=|\xi|$ with $\xi\sim\mathcal{N}(\mu,\sigma^2)$. The exact integral representations for the partition function, internal energy, free energy, heat capacity, and entropy are derived, and analytic approximations are given in two complementary limits—small variance ($\sigma\ll\mu$) via a cumulant expansion, and the zero-center case ($\mu=0$) via low-frequency asymptotic analysis. The model is extended to $N$ independent oscillators, where the heat capacity is shown to be extensive with self-averaging fluctuations $\propto N^{-1/2}$, and finally to a disordered oscillator lattice, where the folded-Gaussian kink at $\omega=0$ produces a soft-mode infrared tail. For a single isolated oscillator with $\mu=0$, both $C$ and $S$ vanish linearly at low $T$. In the lattice case, the van Hove factor converts this to a $T^d$ power law. The oft-quoted “third-law violation” for disordered phonons is here shown to be a spectral property—the absence of an energy gap and a power-law freeze-out—driven by the single-site distribution kink rather than by a genuine Lifshitz tail (which requires rare large-scale spatial fluctuations). The folded Gaussian thus serves as a minimal benchmark for soft-mode disorder thermodynamics.

We investigate the emergence of a scarred discrete time crystal (SDTC) phase in a periodically driven dimerized spin chain. While generic interacting Floquet systems are expected to thermalize according to the eigenstate thermalization hypothesis (ETH), we demonstrate that this system hosts quantum many-body scars (QMBS) that induce a regime of weak ergodicity breaking. Through an analysis of Floquet level statistics, entanglement entropy, and eigenstate fidelity, we identify a manifold of low-entanglement states characterized by semi-Poisson statistics embedded within an otherwise thermal spectrum. These scarred states support robust subharmonic oscillations with period doubling, signaling the spontaneous breaking of discrete time-translation symmetry. We show that the SDTC response is robust against a variety of initial state configurations, demonstrating its stability beyond fine-tuned conditions. A finite-size scaling analysis reveals that the time-crystalline lifetime grows with system size within the range accessible to our exact-diagonalization calculations. However, drawing on the general phenomenology of approximate many-body scars, we expect that hybridization between Floquet scars and the thermal continuum will eventually curtail this growth, causing the lifetime to saturate at system sizes beyond our current numerical reach. This characterizes the SDTC as a long-lived metastable dynamical regime rather than a strictly stable thermodynamic phase, providing a comprehensive framework for understanding the interplay between periodic driving and constrained many-body dynamics in disorder-free systems.

We construct the natural generalization of stochastic quantization (in the Markovian sense) by considering jump-diffusion processes. This class of stochastic processes exhibits non-continuous paths, so-called Lévy flights. In the presence of jumps, action landscapes with barriers can be efficiently explored, improving and even restoring ergodicity where traditional diffusion approaches become inefficient. We explore different strategies for constructing efficient jump updates, which we deploy to address the benchmark problem of topological freezing in 2d U(1) gauge theory.

Adjoint methods provide a powerful route for gradient-based optimization, but their physical implementation is obstructed in generic nonlinear systems because the adjoint dynamics requires backward-time evolution, Jacobian transposition, and terminal-value constraints. Here we show that nonlinear parity-time ($\mathcal{PT}$)-symmetric systems overcome this obstruction. Using a class of nonlinear non-Hermitian resonator networks, we establish symmetry relations that map the formal adjoint dynamics onto experimentally accessible forward-time evolutions supplemented by controlled injections. This construction enables exact in-situ evaluation of adjoint gradients without requiring explicit backward propagation or matrix transposition. We demonstrate the approach in nonlinear $\mathcal{PT}$-symmetric resonator chains, where the resulting optimization protocol autonomously discovers parameter configurations that realize prescribed spatio-temporal functionalities, including uniform energy redistribution and targeted wave transport at predefined time windows. Our results identify $\mathcal{PT}$ symmetry as a resource for implementing computational sensitivities within physical systems and establish a route toward self-optimizing nonlinear machines.

Neural representations have become a central tool for studying the internal mechanisms of modern AI models, yet their complex high-dimensional structure makes them difficult to interpret. We show that classification tasks give rise to a universal representational geometry, shared across state-of-the-art models in vision, audio, and language processing. The key structure is that within-class variability is not random in representation space. Instead, its classifier-relevant component has strong and structured correlations with the class's own centroid and with the centroids of its competing classes. Building on this observation, we derive an analytical mean-field theory governed mainly by the variability along true-class and rival-class centroid coordinates, together with a global renormalization of the class radius that compensates for the non-Gaussian statistics of real representations. The theory accurately predicts classification accuracy across architectures and modalities. The relevant geometric quantities improve systematically with model scale, mirroring the observed gains in accuracy. A striking feature of the theory is its sparsity: accurate prediction requires only a small set of centroid coordinates associated with the true class and its strongest rivals - connecting our framework to sparse-feature extraction approaches such as sparse autoencoders. Together, these results provide a parsimonious predictive theory of neural representations and suggest that classification in deep networks is governed by a sparse, centroid-aligned structure embedded within the full high-dimensional representation space.

Composite media with complex microstructures exhibit highly tailorable mechanical properties but remain challenging to model efficiently and accurately. Conventional homogenization often oversimplifies microstructural effects, whereas multiscale approaches typically require costly coupling across spatial and temporal scales. To address these limitations, we propose a two-scale neural-swarm framework for large-scale mechanical modeling of heterogeneous composites. At the local scale, the mechanical characteristics of representative microstructural features are encoded into building-block Fourier neural operators (FNOs) using level-set representations. At the global scale, these pretrained FNOs are assembled into an FNO swarm according to the spatial distribution of microstructural constituents. A coarse-mesh finite element model is employed to provide global physical guidance, while Schwarz iteration is used to synchronize neighboring FNOs and enforce consistency across shared interfaces. The proposed framework is validated through nonlinear simulations of SiC-Al composites with diverse microstructural configurations. Compared with nonlinear finite element analysis, the FNO-swarm method achieves comparable accuracy while reducing computational cost by orders of magnitude. For an extreme dual-property SiC-Al composite containing more than a billion nodal points, the proposed approach predicts the mechanical response within approximately one hour, demonstrating exceptional scalability. Furthermore, the framework naturally accommodates arbitrary Dirichlet boundary conditions and complex domain geometries. The proposed neural-swarm strategy provides a robust and scalable paradigm for large-scale mechanics simulations, reconciling the longstanding trade-off between computational efficiency and physical fidelity in heterogeneous materials modeling.

A persistent challenge in computational physical modeling is the substantial disparity between the characteristic length scales of microstructures and macroscopic structural components. Multiscale modeling has been widely adopted to bridge this gap by coupling methodologies tailored to different scales. However, conventional approaches, such as asymptotic homogenization (bottom-up) and submodeling (top-down), often entail rigorous mathematical prerequisites or intricate interfacing procedures. To address these limitations, we introduce a fully scalable neural-embedded graphical model (NEGM) that provides a unified framework for the progressive upscaling of highly heterogeneous composite materials. Specifically, NEGM encodes all microstructure- and material-related complexities into constituent neural network blocks, which are then organized into a hypergraph to simulate progressively larger domains. Extensive numerical benchmarks demonstrate that NEGM reliably predicts the physical responses of 2D and 3D composites exhibiting strong material nonlinearity, arbitrary boundary conditions, and irregular geometries. Crucially, because NEGM relies solely on neural network training and inference, it offers a scale-invariant formulation. This enables iterative application of NEGM to upscale from the microscale to arbitrarily large scales, circumventing the need for complex interfacing protocols between disparate modeling frameworks. We validate this progressive upscaling strategy on a large mosaic composite domain, showing that the accumulated error can be effectively contained provided the constituent blocks achieve sufficiently high prediction accuracy. Our findings suggest that artificial neural networks not only enhance the efficiency of direct single-scale simulations, as previously demonstrated, but also provide a clean and elegant pathway toward streamlined multiscale modeling.

We study reflection-resonance poles $Z_j=E_j-i\Gamma_j$, $\Gamma_j>0$, of a finite one-dimensional disordered sample of length $L$, coupled at one end to a semi-infinite lead, in the regime $L\gg\ell_L\gg k^{-1}$, where $\ell_L$ is the localization length and $k=\sqrt{E}$. The key step is to relate the resonance density to the reflection coefficient at the complex energy $E+i\Gamma$, corresponding to uniform absorption. Exact finite-chain Kac–Rice and Poincaré–Lelong identities reduce pole counting to a finite-length diffusion of the reflected intensity. For the perfect contact transparency the density crosses over from the localization-controlled $\Gamma^{-1}$ law to the broad-resonance $\Gamma^{-2}$ law at $\Gamma_L=k/\ell_L$, in agreement with $L\to\infty$ result of Fyodorov and Meibohm. Finite length cuts off the $\Gamma^{-1}$ regime at $\Gamma_{\rm ultra}=\frac{e^{1-\gamma_{\rm E}}}{2}\Gamma_L e^{-L/\ell_L}$. We derive the ultranarrow resonances crossover shape as an explicit moving front; for contact transparency $\mathcal T<1$ this scale shifts to $\mathcal T\Gamma_{\rm ultra}$. The same reflection process yields the finite-length Wigner time-delay statistics and, in the weak-absorption limit, the Comtet–Texier distribution. We show that at eigenvalues of the corresponding closed Dirichlet sample the Wigner delay is inversely proportional to the squared boundary derivative of the normalized eigenfunction. This quantity also gives the eigenvalue response to displacement of the Dirichlet boundary at the lead-contact end, and hence the force exerted by the eigenmode on that boundary; we obtain its finite-length distribution. Finally, the finite-transparency resonance density obeys the single-channel Moldauer–Simonius sum rule, linking its perfect-coupling divergence to the $\Gamma^{-2}$ tail. Direct lattice and spectral computations test the crossover and front constant.

We investigate a one-dimensional non-Hermitian hexagonal Harper model with quasiperiodically modulated hopping amplitudes. In the Hermitian limit, the model exhibits metallic, insulating, and multifractal phases characterized by distinct eigenstate properties. Upon introducing non- Hermiticity, the phase diagram is qualitatively altered, with an expansion of metallic regions and strong boundary sensitivity arising from the non-Hermitian skin effect. By analyzing wave-packet dynamics, we uncover qualitatively distinct transport signatures in metallic, multifractal, and in- sulating regimes. In the metallic region, nonreciprocal hopping induces finite sliding, resulting in ballistic center-of-mass motion that is absent in the Hermitian model, while wave-packet spreading is simultaneously suppressed and exhibits diffusive scaling. Interestingly, the multifractal regime emerges as a distinct dynamical phase supporting both enhanced spreading and finite sliding, both primarily of superdiffusive nature, in contrast to metallic regions where sliding (spreading) shows ballistic (diffusive) scaling. These features are markedly different from their Hermitian counterpart. On the other hand, in the insulating region, both the spreading and sliding are strongly suppressed. We reconfirm these intriguing transport characteristics by investigating the distinct growth profile of single-particle entanglement entropy where the effect of spreading of the wave-packet is clearly manifested. These results demonstrate that quasiperiodicity in hopping amplitudes, combined with non-Hermiticity, establishes the multifractal regime as a key mediator of transport.

Extinction is inevitable; every species eventually dies out, impacting the ecosystem it is part of. Over the past few decades, extensive research stemming from the stability-diversity debate has addressed how species diversity contributes to the stability of large ecosystems. However, conventional stability criteria often rely on deterministic frameworks, overlooking the intrinsic population fluctuations that allow any species to go extinct by chance. In this paper, we incorporate demographic stochasticity into large complex ecosystems with a rule-based model. We demonstrate that such demographic fluctuations rapidly prune the low-abundance species from the ecological community, thereby securing higher systemic stability. By developing a bottom-up theory to characterise the statistics of extinction dynamics, we discover that the fraction of surviving species exhibits an anomalous heavy-tailed decay over time, revealing the emergence of remnant communities with robust metastability. Our results highlight that when demographic fluctuations are accounted for, ecosystems self-stabilise by reducing their diversity even when they are predicted to be chaotic in the deterministic limit.

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<\beta\leq\beta_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<\beta\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<\beta\leq\beta_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erdős-Rényi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.

Complex multipath environments are usually avoided in wave-based information processing because repeated scattering creates many interfering propagation paths, obscuring controllability and generating extreme sensitivity to perturbations. The addition of nonlinear mechanisms fundamentally alters the wave-control landscape by breaking the superposition principle that underpins most wave-management strategies. Here, we show that these two apparent impediments – multipath complexity and nonlinearity – can instead be harnessed as key resources for physical optimization. We demonstrate an in-situ adjoint optimization protocol in a wave-chaotic platform incorporating a single localized nonlinear defect, in which the system itself performs both the forward and the adjoint propagations required for gradient evaluation. Recurrent multipath returns repeatedly expose the wave to the defect, producing from a minimal hardware a rich nonlinear input-output map with many pathway-mediated degrees of freedom. At the same time, a suitable adjoint excitation enables direct extraction of the sensitivities from measurements alone, without a digital twin or conventional numerical backpropagation. We experimentally validate the protocol on a minimal nonlinear multipath platform composed of incommensurate coaxial cables connected via T-junctions, one of which hosts a diode-loaded cavity. Our approach opens a route to adaptive wireless communications, imaging and analog intelligence in complex, partially unknown environments where conventional modeling is impractical.

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single “structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the “order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka–Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

We report the magnetic properties of a series of lithium-rich antiperovskites (Li$_2M$)$Ch$O ($M$ = Fe, Co, Mn and $Ch$ = Se, S) where transition metal and lithium ions are randomly distributed on the X-sites of the X$_3$BA structure, thereby forming a strongly diluted magnetic sublattice. Our study hence enables us to investigate the evolution of magnetic order at fixed 1/3-filling – which is in the vicinity but slightly above the percolation threshold – upon variation of the spin size, the magnetic anisotropy, and the orbital configuration. The data imply the absence of a distinct Curie-Weiss behavior up to 350~K but show rather large and weakly temperature-dependent magnetic susceptibility. We observe clear signatures of long-range antiferromagnetic order evolving in the 1/3-filled and strongly diluted magnetic X-site lattice with increasing Néel temperatures from $T_{\rm{N}}\simeq 30$~K in (Li$_2$Mn)$Ch$O to $\simeq 50$~K in (Li$_2$Fe)$Ch$O and $70-90$~K in (Li$_2$Co)$Ch$O. Except for $M$ = Co, the chalcogenide has no sizable effect on $T_{\rm N}$. We conclude significant magnetic coupling and short-range magnetic correlations at well above $T_{\rm N}$ which is in line with the observation of a broad electron spin resonance signal at room temperature. The actual ordering temperatures are strongly diminished by magnetic dilution. While structural parameters such as the tolerance factor and bonding angles do not strongly affect $T_{\rm N}$, a key parameter is the magnetic anisotropy of the transition metals.

Physical neural networks perform learning through the intrinsic nonlinear dynamics of matter. Optimising their design presents a considerable challenge: complex many-body physics can provide powerful computation, but are expensive to simulate and large experimental optimisation runs are impractical to fabricate. Hence, the high-dimensional space of possible network topologies cannot be effectively directly searched. Here, we show that this search can be efficiently performed in an abstract graph space that is vastly cheaper to explore. Using random lasing networks – composed of interconnected nanoscale waveguides and hosting strongly coupled lasing modes – as an exemplar physical vision system, we establish a quantitative three-layer link: simple graph-theoretic metrics predict the nonlinear lasing physics, which in turn predicts vision performance. After validating this relationship using physical simulations, we exploit it to drive an evolutionary algorithm using graph metrics, producing network topologies that outperform random designs at a fraction of the computational cost (3000$\times$ speed-up compared to physical simulation). On simulated image-classification tasks, graph-optimised networks substantially improve classification accuracy. As our framework operates on network topology rather than substrate-specific physics, we anticipate it can transfer to other network-based physical learning systems, providing an efficient route for the directed design and optimisation of complex, strongly-interacting physical neural networks.