Papers

We study upper large deviations of the free energy of mixed $p$-spin Ising spin glasses through the generalized TAP free energy $F_{\mathrm{TAP}}$ of Chen, Panchenko and Subag. For every mixture with radius of convergence greater than one, $\max F_{\mathrm{TAP}}$ and $\log Z_N$ have the same large deviations at speed $N$; the proof combines their band theorem with Ramsey's theorem. For convex mixtures with $\sum_p 2^p\beta_p^2<\infty$, upper deviations at level $f$ are carried by generalized TAP critical points, which exist at the free-energy rate without Boursier's strict Plefka condition. Outside an event of smaller exponential order, near-maximizers at level $f$ lie near the contact set of a constrained Parisi obstacle, follow the Auffinger-Chen field law, and have Hessian bulk near a reflected Pastur law whose edge is nonpositive and vanishes exactly at contacts where the obstacle is flat to second order. Where every contact is of this kind, a fixed stability margin costs rate.

Let $Z_N$ be the partition function of the Sherrington-Kirkpatrick model at inverse temperature $\beta>1$. Replica theory (Kondor near the critical temperature, Parisi and Rizzo at all temperatures) predicts that the probability that $N^{-1}\log Z_N$ exceeds its typical value by $\delta$ decays like $\exp(-Na\delta^{6/5})$ with an explicit $a$; Aronow and Lopatto recently proved bounds of this order. The exponent is the Legendre transform of the Parisi problem constrained to carry an atom $\theta$ at zero. We show that the constrained minimum exceeds the equilibrium free energy by $c_\beta\theta^5(1+o(1))$, where $c_\beta=\frac{9}{640}\beta^3\rho_\beta(0)^{-3}$ and $\rho_\beta(0)>0$ is the density at zero of the Parisi measure. With the fractional-moment formula this gives the exponent $\frac56(6c_\beta)^{-1/5}\delta^{6/5}(1+o(1))$ as $N\to\infty$ and then $\delta\to0$. The constrained minimizer has a marginal first positive contact, to leading order at $3\theta/(2\rho_\beta(0))$ with an atom $\theta/2$. This plateau calculus applies to every mixture with $\xi'''(0)=0$ whose Parisi measure accumulates at zero; a small field $h$ opens a gap of order $|h|^{2/3}$, and an atom of the Parisi measure at zero, present in every pure $p$-spin model with $p\ge3$, makes the rate leave zero linearly. Near criticality the constant tends to Kondor's value $9/5120$ and the marginal regime ends at a de Almeida-Thouless threshold. For the Sherrington-Kirkpatrick model at every $\beta>1$, a finite-step curvature inequality from an unrefereed preprint shows that every constrained minimizer either has an interval of positive contacts, all marginal, or has exactly two atoms, and that the marginal regime ends at a unique de Almeida-Thouless threshold $\theta_c(\beta)$. The laws at every $\beta>1$ use that the Parisi measure accumulates at zero, a theorem of Lopatto also proved in earlier work of the author.

We investigate the effects of quenched disorder on two-dimensional electron solids using Hartree-Fock theory and classical molecular dynamics. Increasing disorder weakens crystalline correlations, broadens the structure factor, and reduces the measured coherence length toward the pristine lattice spacing. This evolution is consistent with the Imry-Ma and Larkin picture: random pinning disrupts translational order on sufficiently long scales, and stronger disorder reduces the distance over which crystalline organization survives. Classical simulations further show that stronger pinning raises the temperature for particle rearrangement even after crystalline order has disappeared. Disorder thus suppresses structural coherence while stabilizing an amorphous solid against thermal rearrangement.

The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions and uses an especially simple SYK analysis based on a comparison to classical dynamics.

Sompolinsky and Zippelius obtained Parisi's solution of the Sherrington-Kirkpatrick (SK) model from the relaxational dynamics of soft spins in continuous time, by assuming a hierarchy of widely separated time scales. Here we carry out the same programme for a discrete-time dynamics in which all Ising spins are updated at once. This dynamics, synchronous Monte Carlo (SyncMC), introduces Gaussian auxiliary fields so that the spins become conditionally independent, and it samples the Boltzmann distribution exactly for any system with symmetric pairwise couplings. We show that, for any such system, the correlation and response functions of SyncMC obey an exact fluctuation-dissipation relation in discrete time in equilibrium. The dynamical mean-field equations of SyncMC for the SK model contain two terms absent in continuous-time dynamics, a self-coupling of each spin and a noise correlated through the couplings; both affect only the fast relaxation. As a result, under the same assumptions as in the Sompolinsky-Zippelius construction (widely separated time scales, and on each slow scale the same relation with a reduced ratio x, as expected in relaxation from random initial conditions), the long-time structure of SyncMC reproduces the Parisi equations, including the Parisi partial differential equation and the equation for the distribution of local fields. Equilibrium simulations with replica exchange for up to N=2048 spins exclude the replica-symmetric distribution of the actual local fields and are consistent with the Parisi solution. The function x(q) estimated from the overlap distribution, and from the response in relaxation from random configurations for up to N=4096 spins, is also consistent with Parisi's.

We study the recently introduced SWAP ensemble for Ising spin-glasses, where the spins obtain varying lengths, with length scale $\Delta \in [0, 2]$. The lengths can be exchanged in the spirit of the SWAP algorithm for glassy poly-disperse hard-sphere systems. Using an annealing schedule, if the annealing is slow enough, ground states of the corresponding Ising Hamiltonian, where the spin lengths are incorporated into the bonds, can be obtained with high probability. We prove this statement for two-dimensional systems with sizes of up to $L^2=1024^2$ spins by comparison with exact ground states obtained by using graph matching algorithms. In particular, we analyze the nature of the obtained realizations of the disorder by calculating exact zero-temperature domain-wall energies, from exact ground state calculations using periodic and anti-periodic boundary conditions. For medium or large values of $\Delta$, e.g. $\Delta=1$, the realizations turn ferromagnetic if the annealing is slow enough, i.e., they lose their zero-temperature spin-glass property. If $\Delta$ is small, the realizations remain glassy, but the SWAP annealing does not easily find a true ground state.

Nonlinearity can induce topological phase transitions in periodic systems. It is hence of considerable interest to examine if such transitions persist in the absence of translational symmetry. We address this question directly in real space, using the periodic Su-Schrieffer-Heeger (SSH) chain as a reference for understanding the effect of quasiperiodicity. How nonlinearity drives a chain towards a topological phase transition is found to depend strongly on the strength of the quasiperiodic potential. As the nonlinearity strength increases, we find that weak quasiperiodicity supports a transition from a trivial to a topological phase, whereas the intermediate regime does not have a topological phase transition. Under strong quasiperiodicity, pronounced boundary localization can develop well before a nonlinearity-induced topological transition. Our results show how nonlinearity may interplay with a quasiperiodic potential in inducing topological phase transitions and that boundary localization of nonlinear eigenstates alone may not be taken as a clear signature of topological edge states.

We generalize the Furusawa–Kaneko model, a simple model of intracellular catalytic reaction networks, to multicatalyst reactions and analyze the generalized model using dynamical mean-field theory. In the thermodynamic and dense-network limit, we derive exact effective equations and analyze their fixed points for arbitrary degree distributions. Under nutrient-poor conditions, network heterogeneity suppresses the metabolic–starvation transition irrespective of the number of catalysts per reaction. In contrast, under nutrient-rich conditions, the bifurcation structure changes with the number of catalysts. With one catalyst per reaction, only continuous transitions occur, whereas two or more catalysts give rise to bistable phases and discontinuous transitions. The bifurcation structure also differs between two and three catalysts per reaction, with three catalysts allowing bistability between metabolic states with different cell growth rates. Our results provide a possible mechanism for bistability and abrupt switching in metabolic dynamics.

We show that $\mathcal{PT}$-symmetry acts as the effective time-reversal symmetry governing Anderson localization in the antiferromagnet BaMn$_2$Bi$_2$, whereas the general time-reversal symmetry is broken by its long-range magnetic order. The magnetoconductance follows the laws established from breaking time-reversal symmetry throuhg the orbital coupling of charge carriers to the magnet vector potential, however, the degrees of symmetry breaking are qualitatively modulated by the magnetocrystalline anisotropy stabilizing $\mathcal{PT}$-related sublattices. The quantum interferences and hoping amplitudes governing the phenomena are drastically impaired by transverse magnetic fields, which readily cant $\mathcal{PT}$-related magnetic sublattices. Accordingly, when the $\mathcal{PT}$-related texture is weakly perturbed by longitudinal fields orienting along the sublattices, the localization is minutely affected. The robustness of the $\mathcal{PT}$-enforced degeneracies is thus governed by exchange and magnetocrystalline interactions, in contrast to the exact $\mathcal{PT}$-symmetry of the full Hamiltonian, which is broken by any finite field.

Ising machines have emerged as promising hardware accelerators for intractable optimization and sampling problems, yet their practical impact increasingly hinges on the co-design of algorithms and hardware, where algorithmic demands shape new architectures and new hardware capabilities inspire entirely new algorithms. In this Review, we survey probabilistic algorithms designed for portability across diverse Ising platforms, advocating a top-down perspective that prioritizes principled methods with provable guarantees. We cover foundational methods such as simulated annealing and parallel tempering, including two-dimensional extensions that natively encode hard constraints, and examine approaches that expand the scale of solvable problems from cluster mean-field methods to variational samplers. We highlight the Probabilistic Approximate Optimization Algorithm (PAOA), a classical analog of QAOA that emerged directly from probabilistic hardware development, and explore how generative AI and Ising machines might reinforce each other: learned models propose global moves to accelerate optimization, while probabilistic techniques improve inference in large language models. Much as quantum computing has seen algorithms co-evolve with hardware, probabilistic and Ising computing stand at a similar inflection point. We outline a co-design framework for accelerating the capabilities and adoption of next-generation Ising machines.

Griffiths phases provide a mechanism for sustaining critical dynamics over extended parameter regions without fine-tuning to an isolated transition point. This behavior has been mainly associated with structural heterogeneity in complex networks. Here we show that quenched heterogeneity in local parameters can generate a chaotic Griffiths phase in a network of map-based neurons. We study globally coupled Chialvo maps with quenched disorder in a local neuronal parameter, thereby isolating intrinsic parameter disorder from topological disorder. As the coupling strength is varied, a chaotic Griffiths phase emerges between desynchronized and coherent chaotic dynamics, characterized by intermittent formation and breakup of neuronal clusters and large fluctuations in the fraction of clustered neurons. Within this regime, cluster sizes exhibit power-law distributions with coupling-dependent exponents, providing evidence of critical behavior over a finite parameter interval. The chaotic Griffiths phase is absent in the homogeneous limit and broadens progressively as the degree of parameter heterogeneity increases. We further introduce topological disorder through small-world connectivity and find that its combined action with local parameter heterogeneity substantially extends the chaotic Griffiths phase. In both globally coupled and small-world networks, the number of positive Lyapunov exponents displays anomalous scaling with system size. These results demonstrate that intrinsic dynamical heterogeneity constitutes an independent mechanism for generating chaotic Griffiths phases and that its interplay with structural disorder enhances extended criticality in neuronal networks.

Topological Deep Learning (TDL) is designed to learn features associated with higher-dimensional simplices including not only nodes but also edges, triangles and so on. However, most TDL approaches, being based on boundary operators and Hodge Laplacians, have the limitation that features and signals cannot be lifted or lowered across more than one dimension per layer. To overcome this limitation, in this work, we propose the adoption of the cup and cap products. Specifically, we formulate the Cup and Cap Topological Neural Network (CCNN), a topological deep learning architecture designed to learn node-based variables (or 0-cochains) by taking into account their many-body interactions (e.g. triangles) present in the data. We validate CCNN by assessing its performance on the TopoBench datasets, revealing its competitiveness with respect to other simplicial complex neural networks.

The spectral fluctuations in complex quantum systems are modeled in terms of spectra from random matrices. Quantitative agreement with random matrices holds only when the quantum spectrum is a pure sequence, and not a composite spectrum that mixes levels from different symmetry sectors. It was shown earlier that discrete symmetries in composite spectra can be detected using higher-order spacing ratio statistics provided the symmetry sectors are of equal dimensions. The general case is when the symmetry sectors have unequal dimensions, and the symmetries of the Hamiltonian system are unknown or only partially known. In this work, we address how the number of symmetry sectors can be determined from a composite spectrum arising in Hermitian and non-Hermitian settings. A Best Subset Technique is proposed for determining the number of the symmetry sectors by comparing the spectral fluctuations to that of an appropriate random matrix ensemble without requiring de-symmetrization. It is demonstrated using spectra from Gaussian, Wishart and Ginibre ensembles, and also for spectra drawn from many-body quantum systems.

The landscape of topological insulators has expanded beyond its traditional domain of periodic lattices with short-range hopping. Related studies have explored topological phenomena under non-Euclidean geometries, disordered structures, and long-range hopping, progressively narrowing the gap between topological phases of matter and complex networks. Here we demonstrate that genuine complex networks, far beyond periodic lattices and their conventional variants, can themselves host topological insulating phases. By developing a nonconflicting design framework for multiple topological objectives, we realize topological insulators in high-degree Hofstadter lattices and across regular, small-world, and random networks. This generalization uncovers distinct network-specific features, including an expanded range of attainable topological characteristics, a degree-dependent transition from Hofstadter-type to Haldane-type Chern insulators, and, most notably, the superior robustness and reconfigurability of small-world topological insulators. Our graph-theoretic framework establishes network complexity as a resource for realizing high-capacity and reconfigurable topological insulators.

Disordered stealthy hyperuniform (SHU) systems are emerging as platforms for controlling classical and quantum transport due to vanishing single-scattering contributions in the stealthy regime. Here, to our knowledge, we present the first perturbation theory of one-dimensional quantum transport through a continuum of identical point scatterers with SHU positional disorder. Our perturbation theory indicates the exact cancellation of all the lower-order terms due to the Fourier-space stealthy condition $S(q)=0$, and the further suppression of the lowest non-vanishing fourth order in the Lyapunov exponent $\lambda(k)$ due to the real-space hyperuniform constraint $\lim\limits_{D\to\infty}\sigma_D^2=c<\infty$. Our findings presents universal implications of the general wave-propagation phenomena in disordered SHU systems.

Shear transformations are the elementary events of plasticity in glasses and underpin many mesoscopic models of plastic flow, yet their atomic origin and energetics remain difficult to establish. Here we use a machine-learned interatomic potential to show that, in amorphous carbon as a representative network glass, a shear transformation corresponds to the breaking or formation of a single covalent bond. The simplicity of the associated reaction coordinate, the bond length, enables us to extract energy landscapes and barriers. Despite the intrinsic structural disorder, we find that all energy landscapes collapse onto a universal barrier shape. What remains bond-specific is a single scale, the bond-jump distance, which fixes the barrier height through a cubic law. Combined with the activation statistics of shear transformations, this yields a model that predicts the plastic flow stress from independently determined bond-level quantities. Our results provide microscopic support for phenomenological models of plastic flow in glasses and suggest a route towards a quantitative description of amorphous plasticity.

Physical aging in glasses is often described using the concept of material time, which assumes the underlying relaxation mechanism to remain the same as in equilibrium, but with a rate that changes during aging. Whether material time also captures the microscopic dynamics of metallic glasses, which differ qualitatively between the glass and the equilibrium supercooled liquid, is unclear. Using simultaneous time-resolved X-ray diffraction and X-ray photon-correlation spectroscopy, we follow the full equilibration of a metallic glass after small temperature jumps starting from equilibrium. The time evolution of the first sharp diffraction peak position exhibits the asymmetry between temperature up- and down-jumps characteristic of material-time aging and is quantitatively described by the Tool-Narayanaswamy-Moynihan model, which is based on this concept. By contrast, the microscopic density-density autocorrelation functions strongly violate material-time scaling. Compressed-exponential relaxation emerges even after temperature changes as small as 4K, irrespective of the direction of the temperature jump, and continuously evolves into a stretched-exponential relaxation shape following the equilibration of the average structure. This behavior is consistent with a transient stress-mediated contribution associated with volumetric frustration that disappears upon equilibration. Our results show that a material time can describe the evolution of average structural properties even when the corresponding microscopic dynamics do not obey material-time scaling.

Nearest-neighbor (NN) tight-binding models (TBMs) on plaquette-centered $\{ 10,3\}$ (Schläfli symbol), $\{ 8,3\}$, and $\{ 8,4\}$ hyperbolic lattices on a Poincaré disk with open boundary conditions display a vanishing, a finite, and a diverging density of states (DOS) near zero energy (band center), respectively, yielding a Dirac liquid, a Fermi liquid, and a flat band. Emergent bipartite nature of these lattices within the framework of NN-TBMs, allows us to scrutinize the impact of mass disorder on the electronic states and DOS therein. From extensive numerical calculations of the average and typical DOS using the kernel polynomial method, we show that hyperbolic Dirac liquid remains stable against weak mass disorder, undergoing a semimetal-to-metal quantum phase transition at moderate disorder, followed by an Anderson metal-to-insulator transition at even stronger disorder. The remaining two systems display only the latter transition. Critical exponents near all these transitions are found to be close to the ones mediated by on-site potential disorder.

We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $\Omega(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

In-context learning (ICL) enables a pretrained model to infer a task from demonstrations without updating its parameters. While much of the existing theory focuses on linear target functions, in this paper we study nonlinear cases by comparing two one-layer attention architectures on the same family of single-index tasks. A kernel learner first maps inputs through a fixed nonlinear feature map and then applies linear attention, whereas a feature learner applies attention to the original input, followed by a learned nonlinear readout. We derive predictions for their memorization and generalization errors using the replica method, retaining the effects of pretraining size, task-pool diversity, and training and inference context lengths. The resulting predictions closely match numerical experiments across a broad range of regimes. Our analysis yields phase diagrams that characterize when each architecture is advantageous as the amount of pretraining data, task diversity, and context lengths vary. We further identify qualitatively different context-length scalings for the two learners. Together, these results clarify how architectural choices interact with the dataset and govern nonlinear in-context learning.

Dense melts of nonconcatenated ring polymers exhibit anomalous dynamics and nonlinear rheology, in some cases linked to long-lived inter-ring threading entanglement. Detecting these geometric constraints remains computationally demanding, with different methods being based on different geometric features tuned to specific models. Here, we introduce a machine-learning framework that detects threading directly from the writhe of ring conformations and we demonstrate its broad generalisability. Trained on simulations of isolated, distance-constrained pairs of short rings, a one-dimensional convolutional neural network (CNN1D) operating on the writhe profile classifies threading states with over 98% accuracy on held-out ring pairs. Remarkably, this model generalises without retraining to equilibrium dense melts spanning a wide range of densities and chain lengths (N = 100 to 1600), maintaining true-positive rates above 91% with false-positive rates near zero. Misclassified conformations are limited to physically ambiguous, shallow threading events which are arguably not impacting the dynamics of the rings. Effectively, writhe captures the interaction between the two rings, allowing writhe-based classifiers to improve the topological analysis of large ring-polymer systems with a 3-fold speed-up over the state-of-the-art minimal-surface detection. These results establish the writhe-trained neural networks as an accurate, scalable tool for characterising entanglement in ring polymer melts.

Why can masked prediction learn useful representations that unmasked reconstruction misses? We study this question in a high-dimensional model of a masked autoencoder (MAE) trained on data with shared latent structure and heterogeneous noise. We prove that masked linear reconstruction can recover the latent feature at linear sample complexity in regimes where unmasked linear reconstruction, equivalent to PCA, fails. The analysis also quantifies the statistical advantage of mask resampling, an established ingredient of masked pretraining. By introducing a fixed collection of $K$ masks per sample, we characterize its effect on feature recovery and downstream performance, identifying regimes where greater mask diversity lowers sample complexity. Guided by this prediction, we find that random cropping and flipping in standard image-training pipelines can obscure the advantage of mask resampling by renewing the prediction task even when the patch mask is fixed. Removing these transformations reveals a downstream advantage for dynamic over static masking in CNN autoencoders and vision transformers. A complementary BERT pilot finds benefits from greater mask diversity on downstream language tasks. Our results separate the benefit of the masked prediction objective from that of mask diversity, and show how a tractable theory can guide experiments that uncover advantages hidden by standard training practices.

Generative models have made rapid progress in ordered crystal structure prediction, yet many functional materials are intrinsically disordered, with substitutional mixing, vacancies, or interstitial species controlling their properties. Existing crystal generators either assume deterministic site occupations or require site-level disorder annotations, which are often unavailable when the chemical formula is the primary input. We formulate disordered crystal structure prediction through an Occupancy Distribution Matrix (ODM), a continuous site-by-species representation that unifies ordered crystals, solid solutions, vacancy disorder, and interstitial occupancy. A valid ODM must satisfy coupled site-wise occupancy, mass-conservation, and non-negativity constraints, placing each sample on a formula-dependent transportation polytope. We propose Entropic Polytope Flow (EP-Flow), a marginal-constrained flow matching framework that canonicalizes heterogeneous polytopes into a shared double-centered space, learns a marginal-preserving flow, and recovers feasible occupancies through a Sinkhorn inverse map. By jointly generating occupancies, fractional coordinates, and lattice parameters, EP-Flow achieves state-of-the-art performance on formula-conditioned disordered CSP benchmarks derived from COD and MPDS, substantially outperforming adapted ordered-crystal generators. Analyses further show that EP-Flow recovers sparse and chemically meaningful local disorder patterns rather than merely matching global composition statistics.

We conduct a numerical study of a generalization of the Blume-Capel model in which the crystal field $D_i$, which controls the local vacancy density, is site dependent rather than uniform. We employ Markov Chain Monte Carlo to compute the shifted phase transition line $T_c$ as a function of the width $\sigma$ of a gaussian distribution $P(D_i) \sim \exp( -(D_i - \bar{D})^2/2 \sigma^2)$ at fixed $\bar{D}$. We characterize how the presence of this on-site Gaussian disorder reduces $T_c$ across the second order transition line, and discuss how our results suggest that $T_c$ remains finite even at large values of $\sigma$. We also determine the effects of disorder on the tricritical point of the conventional, uniform, Blume-Capel model, finding that it significantly suppresses the first-order phase transition, eliminating it entirely at large values of $\sigma$.

On cooling liquids towards their glass-transition temperature, the dramatic increase of their relaxation times is accompanied by changes in particle dynamics at two distinct temperatures: A high-T crossover, where particles become temporarily caged by their neighbors, and a low-T crossover, where the cage escape mechanism changes. While there is some evidence that the former is associated with a change in local particle arrangement, no structural modification has so far been detected across the low-T crossover, fueling scepticism about the relevance of structure for glassy dynamics. Here, we introduce a novel four-point correlation function which allows to determine a structural length scale characterizing cage anisotropy. Extensive molecular dynamics simulations reveal that this scale, as well as the mean structural length scale of the glass-former, extends into the medium range, i.e., significantly exceeds the particle size. Strikingly, the difference between these two scales - a measure of the degree of cage anisotropy - peaks at both crossover temperatures. We discuss how the presence of these peaks enables understanding the nature of the change in the microscopic transport mechanism at the two temperatures, thereby linking both dynamical crossovers to a single structural observable, the anisotropic medium-range order. This fundamental insight demonstrates that structure, extending well-beyond the local cage, is an essential ingredient for understanding the relaxation dynamics of deeply supercooled liquids.

We investigate fatigue failure in two-dimensional (2D) model glasses under cyclic shear deformation using atomistic simulations. We find that the number of cycles to failure diverges as a power law as the strain amplitude approaches the fatigue limit, with an exponent close to $-1$, in contrast to the exponent of $-2$ reported in three dimensions (3D). A failure exponent of $-1$ has also recently been observed in a 2D elastoplastic model, suggesting an interesting dependence on spatial dimensionality that needs to be rationalized. Measures of accumulated plastic activity, including dissipated work and non-affine displacements, exhibit scaling with the failure time that is consistent with results in 3D, and indicate a robust connection between damage accumulation and failure. To probe the origins of the variability of failure times, we perform isoconfigurational seeded simulations in which a localized soft region is introduced. While such seeding constrains the spatial location of failure, the distribution of failure times remains broad. These findings extend recent results in 3D concerning fatigue failure times to 2D glasses, including their relation to accumulated plasticity and apparent stochasticity, while also revealing a key difference, namely the exponent describing the divergence of the failure times.

Activity can enhance the long-wavelength density fluctuations of a solid beyond the equilibrium fluctuations predicted by the Mermin-Wagner-Hohenberg theorem, but the microscopic origin of this enhancement remains unclear. We study a two-dimensional active Brownian crystal whose longitudinal and transverse displacement covariances scale as $C^u_\lambda(q)\sim q^{-3}$ as compared to the passive counterpart with $q^{-2}$ law. This spectrum predicts an MSD-plateau divergence proportional to $L$ in two dimensions, logarithmic in three dimensions, and finite in four dimensions, in close correspondence with recent results reported in Dey et al. [Nat. Commun. 16, 5498 (2025)]. Inverse of the Covariance gives $H^{\mathrm{cov}}_\lambda\sim |q|^3$ with the corresponding dispersion $\omega_{\mathrm{cov},\lambda}\sim q^{3/2}$, whereas direct mechanical response gives $K_{\mathrm{resp},\lambda}\sim q^2$ and acoustic modes with $\omega_{0,\lambda}\sim q$. The nonlinear dispersion is therefore covariance-defined rather than mechanical. The mobility and integrated active-force correlations remain non-singular, although a finite-persistence crossover cannot be ruled out completely with the existing data. Microscopically, the longitudinal and transverse Irving-Kirkwood interaction-stress spectra scale as $q^{-1}$. The transverse stress-stress correlation connects well with the anomalous displacement field, indicating that activity anomalously populates ordinary acoustic modes, while the conservative interaction network transmits the resulting fluctuations as long-ranged stress. A three-dimensional active FCC crystal exhibits the same paired low-$q$ trends.

Neural quantum states (NQS) have emerged as powerful representations of quantum states with rapidly expanding applications across quantum many-body physics. Yet our understanding of when neural networks can efficiently represent physical quantum states remains limited, in part due to the nonlinear parameterization of NQS, the intricate sign structure of quantum states, and the sensitive dependence on basis choice. We introduce Walshness, a complexity metric of a quantum state which quantifies whether it admits a compact neural-network representation. We rigorously prove that a general quantum many-body state admits an efficient NQS representation if and only if it has bounded Walshness for various NQS architectures. We provide an algorithm for finding the optimal basis by minimizing the energy of a corresponding classical spin model. This allows us to empirically improve the learnability of various physical quantum states, including ground states of the transverse-field Ising model and mixed-field toric code, often reducing infidelity by orders of magnitude, and recovers and generalizes the basis choice given by the well-known Marshall sign rule in frustrated antiferromagnets. Our results provide an intrinsic complexity metric for quantum states with provable connections to their neural-network representability.

We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $\beta_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=\alpha k$ with fixed $\alpha>1$, both Prange's algorithm and DQI can sample at any inverse temperature $\beta < \tanh^{-1}(1/\alpha)$ for sufficiently large $k$, well beyond the dynamical threshold $\beta_{\rm dyn} \sim \sqrt{2\ln k / (\alpha k)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.

Designing a material with a desired property amounts to solving an inverse problem: finding the couplings of a microscopic Hamiltonian whose ground state exhibits that property. For strongly correlated quantum systems, solving this problem efficiently remains largely out of reach. To address this challenge, we present a general framework for ab initio inverse design based on Foundation Neural-Network Quantum States, a recent approach in which the ground states of a family of Hamiltonians are encoded in a single variational wave function. Because the ansatz depends explicitly on the couplings, any target property is a differentiable function of them, and the search for the right Hamiltonian reduces to gradient-based optimization in coupling space. We apply this approach to search for nonmagnetic phases of frustrated Heisenberg models on the square lattice with an increasing number of free next-nearest-neighbor couplings, aiming to identify new quantum spin liquid candidates. With a single coupling, the method recovers the known nonmagnetic window of the square $J_1$-$J_2$ Heisenberg model, whereas letting the two diagonal couplings vary independently reveals an extended nonmagnetic region connecting the square-lattice and anisotropic-triangular-lattice regimes. In a search space of eight independent couplings within a $2\times2$ unit cell, which contains several paradigmatic frustrated spin models, the optimization spontaneously converges to the $J_1$-$J_2$-$\delta$ Heisenberg model, in which the diagonal couplings alternate between two values on neighboring plaquettes, a model recently proposed in the context of altermagnetism. Finite-size scaling up to $16\times16$ clusters in this optimal model shows that the ground state has no magnetic, dimer, or plaquette order, establishing it as a new quantum spin liquid candidate, distinct from those previously proposed on the square lattice.