Wigner crystals are extremely fragile, which is shown to result from very strong geometric frustration germane to long-range Coulomb interactions. Physically, this is manifested by a very small characteristic energy scale for shear density fluctuations, which are gapless excitations in a translationally invariant system. The presence of disorder, however, breaks translational invariance, thus suppressing gapless excitations and pushing them to higher density. We illustrate this general principle by explicit microscopic model calculations, showing that this mechanism very effectively stabilizes disordered Wigner lattices to much higher temperatures and densities than in the clean limit. On the other hand, we argue that in two dimensions disorder significantly ``smears" the melting transition, producing spatial coexistence of solid-like and liquid-like regions -- just as recently observed in STM experiments. Our results paint a new physical picture for melting of Wigner-Mott solids in two dimensions, corresponding to a Mott-Hubbard model with spatially varying local electronic bandwidth.

Spontaneous symmetry breaking is a cornerstone modern physics, defining a wealth of phenomena in condensed-matter and high-energy physics, and beyond. It requires an infinite number of degrees of freedom, and even then, for continuous symmetries, it only works if the spatial dimension is not too low, following the classic results of Coleman, Hohenberg, Mermin and Wagner. While usually discussed in the context of quantum and statistical field theories, and in particular, effective field theories, there are advantages in addressing the same kind of phenomena on discrete geometric structures rather than conventional manifolds. When the space is discretized into a lattice, a lucid picture of conventional spontaneous symmetry breaking springs up, with the ultraviolet issues of continuum quantum field theory out-of-sight, and the key effect, which is infrared in nature, revealed through elementary harmonic oscillator networks. From there, it is natural to generalize lattices to other graphs/networks. In this setting, the presence of spontaneous symmetry breaking is controlled by fractional generalizations of resistance distance and the Kirchhoff index, and most broadly by the spectral dimension. Predictably, because of richness of discrete geometric structures in comparison with continuous manifolds, a broader array of geometries emerge where spontaneous breaking of continuous symmetries is blocked by large fluctuations.

The 2024 Nobel Prize in Physics was awarded for pioneering contributions at the intersection of artificial neural networks (ANNs) and spin-glass physics, underscoring the profound connections between these fields. The topological similarities between ANNs and Ising-type models, such as the Sherrington-Kirkpatrick model, reveal shared structures that bridge statistical physics and machine learning. In this perspective, we explore how concepts and methods from statistical physics, particularly those related to glassy and disordered systems like spin glasses, are applied to the study and development of ANNs. We discuss the key differences, common features, and deep interconnections between spin glasses and neural networks while highlighting future directions for this interdisciplinary research. Special attention is given to the synergy between spin-glass studies and neural network advancements and the challenges that remain in statistical physics for ANNs. Finally, we examine the transformative role that quantum computing could play in addressing these challenges and propelling this research frontier forward.

Physics-Informed Neural Networks (PINNs) have recently emerged as a promising tool for fluid dynamics, particularly for flow reconstruction and parameter identification. In the context of granular media, accurately estimating rheological parameters remains a major challenge, as it typically requires complex and costly experimental setups. In this work, we propose a PINN-based approach to identify key rheological parameters of granular materials using a simple experiment: the granular column collapse. A proof of concept is presented using synthetic data, where the PINN is trained to infer the flow fields while simultaneously recovering the rheological parameters. Beyond parameter identification, the method also enables reconstruction of the pressure field, which is difficult to access experimentally. The results highlight the potential of PINNs for data-driven rheometry of granular materials and open perspectives for future applications with real experimental data.

We investigate a variational Monte Carlo framework for trapped one-dimensional mixture of spin-$\frac{1}{2}$ fermions using Kolmogorov-Arnold networks (KANs) to construct universal neural-network wavefunction ansätze. The method can, in principle, achieve arbitrary accuracy, limited only by the Monte Carlo sampling and was checked against exact results at sub-percent precision. For attractive interactions, it captures pairing effects, and in the impurity case it agrees with known results. We present a method of systematic transfer learning in the number of network parameters, allowing for efficient training for a target precision. We vastly increase the efficiency of the method by incorporating the short-distance behavior of the wavefunction into the ansätz without biasing the method.

Two-dimensional (2D) Dirac fermions occur ubiquitously in condensed matter systems from topological phases to quantum critical points. Since the advent of topological semimetals, where the dispersion is often tilted around the band crossing where the Dirac fermion can appear, tilt has emerged as a key handle that controls physical properties. We study how tilt affects the transport and spectral properties of tilted 2D Dirac fermions under scalar disorder. Although our spectral analyses always show conformity to appropriate Gaussian ensembles, suggestive of delocalization, the conductivity scaling $g(L)$ shows a surprising richness. For a single Dirac node, relevant for quantum Hall transitions and topological insulator surface states, we find $g(L)\sim a_1\log(L)$ with a tilt-dependent coefficient a_1>0. Interestingly, when the tilt and transport directions are aligned, $a_1$ and hence $g(L)$ shows a spike at the critical point between the type-I and type-II regimes of the Dirac node. For systems with two Dirac nodes with unbroken time-reversal symmetry, pertinent to quasi-2D Dirac materials, we find $g(L)\sim L^{a_1}(\log L)^{a_2}$. However, we find a surprising tension between tilt along and perpendicular to the transport directions. For the former, $a_1$ changes sign as a function of tilt, hinting at a tilt-driven localization-delocalization transition, while a_1<0 for all tilts in the latter case, implying localization. These localized behaviors also reveal tension with the delocalization seen in spectral properties and suggest differing localization tendencies in real and Hilbert spaces. Overall, our work identifies tilt as an essential control parameter that uncovers rich and unconventional transport physics in 2D Dirac materials.

Many-body localization (MBL) is understood theoretically through the existence of an extensive number of local integrals of motion (LIOMs). These conserved quantities are related to the microscopic quantum degrees of freedom that are spatially localized. Here, we present a general framework for constructing exact LIOMs with the desired locality and quantum numbers supplied as input rather than arising as emergent properties. We show that one can express the task of finding LIOMs as an optimization problem. In simple cases, solving this problem amounts to matrix diagonalization, while in more complex settings, it connects to the question of finding classical ground states of spin-glass models. We illustrate our theory using paradigmatic examples of single-particle Anderson localization and MBL in interacting spin chains. These developments unify previous results and reveal intriguing connections among many-body localization, spin-glass physics and constrained optimization problems.

In this work, we investigate the combined effects of Rashba spin-orbit coupling (RSOC) and non-Hermiticity on topological phase transitions in spinful p-wave Kitaev chains. While previous studies have separately examined non-Hermitian (NH) extensions of Kitaev chains and the effects of RSOC in Hermitian systems, the interplay between these two mechanisms remains largely unexplored. We analyze this interplay by considering two distinct types of complex on-site potentials: (i) a uniform gain/loss term and (ii) a complex quasiperiodic potential. We demonstrate that the impact of RSOC is highly model-dependent. In particular, RSOC does not affect the topological phase boundary in the Hermitian limit of the uniform gain/loss model (provided the spin-flip hopping is weaker than the pairing strength), but significantly alters the topological landscape in the NH regime. In contrast, for the quasiperiodic model, RSOC modifies the phase boundaries in both the Hermitian and non-Hermitian cases. Notably, we find that the combined interplay of non-Hermiticity and RSOC drives topological transitions at significantly lower potential strengths compared to the Hermitian limit. We derive analytical expressions for the topological phase transitions in both cases and validate our predictions through numerical calculations of energy spectra and real-space winding numbers. This work provides a comprehensive understanding of how non-Hermiticity and RSOC cooperatively reshape topological phase diagrams in one-dimensional superconducting systems.

Biological neural networks learn complex behaviors from sparse, delayed feedback using local synaptic plasticity, yet the mechanisms enabling structured credit assignment remain elusive. In contrast, artificial recurrent networks solving similar tasks typically rely on biologically implausible global learning rules or hand-crafted local updates. The space of local plasticity rules capable of supporting learning from delayed reinforcement remains largely unexplored. Here, we present a meta-learning framework that discovers local learning rules for structured credit assignment in recurrent networks trained with sparse feedback. Our approach interleaves local neo-Hebbian-like updates during task execution with an outer loop that optimizes plasticity parameters via \textbf{tangent-propagation through learning}. The resulting three-factor learning rules enable long-timescale credit assignment using only local information and delayed rewards, offering new insights into biologically grounded mechanisms for learning in recurrent circuits.

Quantum spin liquids are elusive long-range entangled states. Motivated by experiments in Rydberg quantum simulators, recent excitement has centered on the possibility of dynamically preparing a state with quantum spin liquid correlation even when the ground state phase diagram does not exhibit such a topological phase. Understanding the microscopic nature of such quantum spin "lake" states and their relationship to equilibrium spin liquid order remains an essential question. Here, we extend the use of approximately symmetric neural quantum states for real-time evolution and directly simulate the dynamical preparation in systems of up to $N=384$ atoms. We analyze a variety of spin liquid diagnostics as a function of the preparation protocol and optimize the extent of the quantum spin lake thus obtained. In the optimal case, the prepared state shows spin-liquid properties extending over half the system size, with a topological entanglement entropy plateauing close to $\gamma = \ln 2$. We extract two physical length scales $\lambda$ and $\xi$ which constrain the extent of the quantum spin lake $\ell$ from above and below.

Recent studies of delocalization-localization transitions in disordered quantum chains have highlighted the role of rare, chain-breaking events that favor localization, in particular for high-energy eigenstates related to many-body localization. In this context, we revisit the random-field XXZ spin-1/2 chain at zero temperature with ferromagnetic interactions, equivalent to interacting fermions or hard-core bosons in a random potential with attractive interactions. We argue that localization in this model can be characterized by chain-breaking events, which are probed by the extreme values of simple local observables, such as the on-site density or the local magnetization, that are readily accessible in both experiments and numerical simulations. Adopting a bosonic language, we study the disorder-induced Berezinskii-Kosterlitz-Thouless (BKT) quantum phase transition from superfluid (SF) to Bose glass (BG), and focus on the strong disorder regime where localization is driven by weak links. Based on high-precision density matrix renormalization group simulations, we numerically show that extreme local densities accurately capture the BKT transition, even for relatively short chains ranging from a few dozen to a hundred sites. We also discuss the SF-BG transition in the weak disorder regime, where finite-size effects pose greater challenges. Overall, our work seeks to establish a solid foundation for using extreme statistics of local observables, such as density, to probe delocalization-localization transitions in disordered quantum chains, both in the ground state and at high energy.

Energy-based models have become a central paradigm for understanding computation and stability in both theoretical neuroscience and machine learning. However, the energetic framework typically relies on symmetry in synaptic or weight matrices - a constraint that excludes biologically realistic systems such as excitatory-inhibitory (E-I) networks. When symmetry is relaxed, the classical notion of a global energy landscape fails, leaving the dynamics of asymmetric neural systems conceptually unanchored. In this work, we extend the energetic framework to asymmetric firing rate networks, revealing an underlying game-theoretic structure for the neural dynamics in which each neuron is an agent that seeks to minimize its own energy. In addition, we exploit rigorous stability principles from network theory to study regulation and balancing of neural activity in E-I networks. We combine the novel game-energetic interpretation and the stability results to revisit standard frameworks in theoretical neuroscience, such as the Wilson-Cowan and lateral inhibition models. These insights allow us to study cortical columns of lateral inhibition microcircuits as contrast enhancer - with the ability to selectively sharpen subtle differences in the environment through hierarchical excitation-inhibition interplay. Our results bridge energetic and game-theoretic views of neural computation, offering a pathway toward the systematic engineering of biologically grounded, dynamically stable neural architectures.

The atomic-scale influence of disorder on the topological order can be quantified by a universal topological marker, although the practical calculation of the marker becomes numerically very costly in higher dimensions. We propose that for any symmetry class in higher dimensions, the topological marker can be calculated in a very efficient way by adopting the kernel polynomial method. Using class AII in three dimensions as an example, which is relevant to realistic topological insulators like Bi2Se3 and Bi2Te3, this method reveals the criteria for the invariance of topological order in the presence of disorder, as well as the possibility of a smooth cross over between two topological phases caused by disorder. In addition, the significantly enlarged system size in the numerical calculation implies that this method is capable of capturing the quantum criticality much closer to topological phase transitions, as demonstrated by a nonlocal topological marker.

We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time $T$ and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing $T$, from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in $d+1$ space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the ``localization'' time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a $d=2$ model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.

Typically, scaling up the size of a system does not change the shape of its energy spectrum, other than making it denser. Exceptions, however, occur in the new phenomenon of non-Hermitian skin criticality, where closely competing generalized Brillouin zone (GBZ) solutions for non-Hermitian state accumulation give rise to anomalously scaling complex spectra. In this work, we discover that such non-Hermitian criticality can generically emerge from stochasticity in the lattice bond orientation, a surprising phenomenon only possible in 2D or beyond. Marked by system size-dependent amplification rate, it can be physically traced to the proliferation of feedback loops arising from excess local non-Hermitian skin effect (NHSE) accumulation induced by structural disorder. While weak disorder weakens the amplification as intuitively anticipated, stronger disorder enigmatically strengthens the amplification almost universally, scaling distinctly from conventional critical system. By representing cascades of local excess NHSE as ensembles of effectively coupled chains, we analytically derived a critical GBZ that predicts how state amplification scales with the system size and disorder strength, highly consistent with empirical observations. Our new mechanism for disordered-facilitated amplification applies generically to structurally perturbed non-Hermitian lattices with broken reciprocity, and would likely find applications in non-Hermitian sensing through various experimentally mature meta-material platforms.

Replica symmetry breaking (RSB) underlies the complex organization of disordered systems, yet quantitative validation beyond $N \sim 100$ spins has remained computationally challenging. We use quantum annealing to access ground states of the Sherrington-Kirkpatrick model up to $N = 4000$ spins, enabling the most extensive test of Parisi's Nobel Prize-winning RSB solution to date. Five independent observables confirm RSB predictions: ground-state energies converge to Parisi's value with characteristic $N^{-2/3}$ corrections, energy fluctuations scale as $N^{-3/4}$ ($\gamma = 0.739 \pm 0.036$), the chaos exponent $\theta = 0.51 \pm 0.02$ ($R^2 = 0.989$) confirms mean-field universality, the overlap distribution exhibits hierarchical structure ($\sigma_q = 0.19$), and the complexity remains invariant under 36\% network dilution. Beyond a critical threshold 0.8 < D_c < 0.9, the hierarchy collapses discontinuously through a cooperative avalanche that converts the entire system to vacancies within a narrow parameter window $\Delta D = 0.1$. These findings establish quantum computation as a tool for probing emergent many-body phenomena and uncover the topological foundations of complexity in disordered systems, with implications for neural networks, optimization, and materials science.

Spin glass systems as lattices of disordered magnets with random interactions have important implications within the theory of magnetization and applications to a wide-range of hard combinatorial optimization problems. Nevertheless, despite sustained efforts, algorithms that attain both high accuracy and efficiency remain elusive. Due to their topologies being low-$k$-partite such systems are well suited to a probabilistic computing (PC) approach using probabilistic bits (P-bits). Here we present complex spin glass topologies solved on a simulated PC realization of an Ising machine. First, we considered a number of three dimensional Edwards-Anderson cubic spin-glasses randomly generated as well as found in the literature as a benchmark. Second, biclique topologies were identified as a likely candidate for a comparative advantage compared to other state-of-the-art techniques, with a range of sizes simulated. We find that the number of iterations necessary to find solutions of a given quality has constant scaling with system size past a saturation point if one assumes perfect parallelization of the hardware. Therefore a PC architecture can trade the computational depth of other methods for parallelized width by connecting a number of P-bits that scales linearly in system size. This constant scaling is shown to persist across a number of solution qualities, up to a certain limit beyond which resource constraints limited further investigation. The saturation point varies between topologies and qualities and becomes exponentially hard in the limit of finding the ground truth. Furthermore we demonstrate that our PC architecture can solve spin-glass topologies to the same quality as the most advanced quantum annealer in minutes, making modest assumptions about their implementation on hardware.

The boundary between classically simulable and computationally superior quantum systems is fundamental to identifying true quantum advantage. We investigate this within the framework of quantum reservoir computing by introducing a tunable $N$-qubit random circuit model, where a fraction $p$ of Clifford gates are probabilistically substituted with nonstabilizing conditional-$\hat{T}$ gates. We establish a direct correspondence between the reservoir's performance on temporal processing tasks and its entanglement spectrum statistics and long-range nonstabilizer resource content. To assess scalability, we study the scaling of the anti-flatness of states in the large-$N$ limit at a fixed circuit depth ratio $d/N \sim \mathcal{O}(1)$. This is taken as a witness to concentration of measures, a known impediment to learning in thermalizing systems. We demonstrate that the learnability and scalability of the reservoir can be continuously controlled by the parameter $p$, allowing us to navigate from classically tractable to maximally expressive quantum dynamics. These architecture-agnostic results offer a general strategy for designing powerful and trainable quantum machine learning systems and clarify the physical resources underpinning quantum computational advantage.

Neural scaling laws underlie many of the recent advances in deep learning, yet their theoretical understanding remains largely confined to linear models. In this work, we present a systematic analysis of scaling laws for quadratic and diagonal neural networks in the feature learning regime. Leveraging connections with matrix compressed sensing and LASSO, we derive a detailed phase diagram for the scaling exponents of the excess risk as a function of sample complexity and weight decay. This analysis uncovers crossovers between distinct scaling regimes and plateau behaviors, mirroring phenomena widely reported in the empirical neural scaling literature. Furthermore, we establish a precise link between these regimes and the spectral properties of the trained network weights, which we characterize in detail. As a consequence, we provide a theoretical validation of recent empirical observations connecting the emergence of power-law tails in the weight spectrum with network generalization performance, yielding an interpretation from first principles.

Training large language models with Reinforcement Learning with Verifiable Rewards (RLVR) exhibits a set of distinctive and puzzling behaviors that remain poorly understood, including a two-stage learning curve, a V-shaped response-length trajectory, and a pronounced vulnerability to catastrophic forgetting. In this work, we propose that these behaviors are emergent collective phenomena governed not by neural implementation details, but by the topological evolution of the latent reasoning graph in semantic space. By demonstrating a dynamical isomorphism between a 1.5B-parameter LLM and a minimal Concept Network Model (CoNet), we trace the causal source to the self-organization of a sparse concept web pinned to an average degree of two. This geometric perspective provides a unified physical explanation for the observed anomalies: the V-shaped trajectory tracks the evolution from parallel local skill optimization to global network integration; catastrophic forgetting stems from the topological disconnection of critical ``trunk'' edges; and policy collapse arises from the accumulation of sequential transitions at the web's leaf nodes, where broad exploration abruptly freezes into rigid, high-reward trajectories. Identifying a ``maximally frustrated state'' at the transition between learning stages, we propose Annealed-RLVR, a principled algorithm that injects a targeted SFT ``heating'' step to resolve this topological bottleneck. Experiments confirm that this theory-driven intervention outperforms standard RLVR on both in-distribution and out-of-distribution benchmarks (including Minerva and AIME). By recasting RLVR from black-box optimization into a predictable process of structural self-organization, our work provides a new physical intuition for engineering the emergent reasoning capabilities of future AI systems.