Phase Transitions

We develop the spectral edge analysis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme aspect ratio regime (parameters $P \sim 10^8$, window $W \sim 10$), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at position $k^* = \mathrm{argmax}\, σ_j/σ_{j+1}$. From three assumptions we derive: (i) gap dynamics governed by a Dyson-type ODE with curvature asymmetry, damping, and gradient driving; (ii) a spectral loss decomposition linking each mode's learning contribution to its Davis--Kahan stability coefficient; (iii) the Gap Maximality Principle, showing that $k^*$ is the unique dynamically privileged position -- its collapse is the only one that disrupts learning, and it sustains itself through an $α$-feedback loop requiring no assumption on the optimizer. The adiabatic parameter $\mathcal{A} = \|ΔG\|_F / (η\, g^2)$ controls circuit stability: $\mathcal{A} \ll 1$ (plateau), $\mathcal{A} \sim 1$ (phase transition), $\mathcal{A} \gg 1$ (forgetting). Tested across six model families (150K--124M parameters): gap dynamics precede every grokking event (24/24 with weight decay, 1/24 without), the gap position is optimizer-dependent (Muon: $k^*=1$, AdamW: $k^*=2$ on the same model), and 19/20 quantitative predictions are confirmed. The framework is consistent with the edge of stability, Tensor Programs, Dyson Brownian motion, the Lottery Ticket Hypothesis, and neural scaling laws.

Grokking -- the delayed onset of generalization after early memorization -- is often described with phase-transition language, but that claim has lacked falsifiable finite-size inputs. Here we supply those inputs by treating the group order $p$ of $\mathbb{Z}_p$ as an admissible extensive variable and a held-out spectral head-tail contrast as a representation-level order parameter, then apply a condensed-matter-style diagnostic chain to coarse-grid sweeps and a dense near-critical addition audit. Binder-like crossings reveal a shared finite-size boundary, and susceptibility comparison strongly disfavors a smooth-crossover interpretation ($Δ\mathrm{AIC}=16.8$ in the near-critical audit). Phase-transition language in grokking can therefore be tested as a quantitative finite-size claim rather than invoked as analogy alone, although the transition order remains unresolved at present.

We study entrywise scalar quantization of two matrices prior to multiplication. Given $A\in R^{m\times k}$ and $B\in R^{k\times n}$, we quantize entries of $A$ and $B$ independently using scalar quantizers with $K_X$ and $K_Y$ levels per entry, and form $\widehat C=\widehat A\,\widehat B$. The objective is to minimize the matrix multiplication mean-squared error (MSE) $E[\|{AB-\widehat A\widehat B}\|_F^2]$ under a pair-i.i.d.\ inner-product model. In the high-resolution regime $K_X,K_Y\to\infty$, we derive a sharp $K^{-2}$ asymptotic expansion for $\mathcal{E}$, identify the exact optimal leading constants, and characterize asymptotically optimal quantization center densities in terms of conditional second moments. We then specialize to correlated Gaussian multiplicative pairs, obtaining a closed-form optimal point density $$λ^\star(u)\ \propto\ \exp\!\left(-\frac{u^2}{6}\right)\bigl((1-ρ^2)+ρ^2u^2\bigr)^{1/3}, \qquad u=\frac{x}{σ_X},$$ with the same form for $y/σ_Y$, and prove a correlation-driven phase transition: the density is unimodal at the origin for $|ρ|\leq 1/\sqrt{3}$ and becomes bimodal for $|ρ|>1/\sqrt{3}$ with peaks at $u_{\mathrm{peak}}=\pm\sqrt{3-1/ρ^2}$. We show our method's applicability in synthetic experiments such as matrix multiplication quantization and least squares optimization, as well as quantization of large language model key and query activations.

Na-ion batteries have emerged as viable candidates for large-scale energy storage applica- tions due to resource abundance and cost advantages. The constraints imposed on their performance and durability, for instance, by complex phase transformations in positive electrode materials during electrochemical cycling, can be addressed and are thus not detrimental to their development. However, diffusion-limited Na-ion transport can drive spatially heterogeneous phase nucleation and propagation, leading to multiphase coexis- tence and locally non-uniform electrochemical activity, generating complex reaction path- ways that challenge both mechanistic understanding and predictive material optimization. These challenges can be addressed by investigating single-crystalline regions of materials, i.e. down to the scale of individual particles, although such analyses are often constrained by energetically and/or spatially sparse hyperspectral datasets. Here, we developed an AI-driven method to process hyperspectral data under sparse sampling conditions and generate multiphase maps with nanometer-scale resolution over a micrometer-scale field of view. We applied this processing on scanning transmission X-ray microscopy (STXM) data to determine the distribution and coexistence of phases in individual particles of NaxV2(PO4)2F3 cathode materials, at different states of charge. The methodology relies on a workflow which combines a Gaussian mixture variational autoencoder (GMVAE) algorithm with the Pearson corre- lation coefficient to identify the sodium content and map their spatial distribution. Our approach reveals nanoscale phase heterogeneity and evolution within individual particles, and improves the reliability of phase detection by identifying ambiguity zones, false assign- ments, and transition phases localized at grain boundaries.

The spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice exhibits a debated intermediate phase between Néel antiferromagnetic and stripe ordered regimes, with competing theories proposing plaquette valence bond, nematic, and quantum spin liquid ground states. We apply the Prometheus variational autoencoder framework -- previously applied to classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the $J_1$-$J_2$ phase diagram using a multi-scale approach. For $L=4$, we employ exact diagonalization with full wavefunction analysis via quantum-aware VAE. For larger systems ($L=6, 8$), we introduce a reduced density matrix (RDM) based methodology using DMRG ground states, enabling scaling beyond the exponential barrier of full Hilbert space representation. Through dense parameter scans of $J_2/J_1 \in [0, 1]$ and comprehensive latent space analysis, we identify the structure factor $S(π,π)$ and $S(π,0)$ as the dominant order parameters discovered by the VAE, with correlations exceeding $|r| > 0.97$. The RDM-VAE approach successfully captures the Néel-to-stripe crossover near $J_2/J_1 \approx 0.5$--$0.6$, demonstrating that local quantum correlations encoded in reduced density matrices contain sufficient information for unsupervised phase discovery. This work establishes a scalable pathway for applying machine learning to frustrated quantum systems where full wavefunction access is computationally prohibitive.

We extend the Prometheus framework for unsupervised phase transition discovery from two-dimensional classical systems to three-dimensional classical systems and quantum many-body systems. Building upon preliminary observations from a 2D Ising model student abstract [Yee et al., 2026], we address two fundamental questions: (1) Does the framework scale to higher dimensions where exact solutions are unavailable? (2) Can it generalize to quantum phase transitions driven by quantum fluctuations rather than thermal fluctuations? For the 3D Ising model on lattices up to $L{=}32$, we achieve critical temperature detection within 0.01% of literature values ($\Tc/J = 4.511 \pm 0.005$) and extract critical exponents with ${\geq}70\%$ accuracy, with statistical analysis correctly identifying the 3D Ising universality class ($p = 0.72$). For quantum systems, we develop quantum-aware VAE (Q-VAE) architectures operating on complex-valued wavefunctions with fidelity-based loss functions, achieving 2% accuracy in quantum critical point detection for the transverse field Ising model. For the disordered TFIM, we perform a consistency check of activated dynamical scaling $\ln ξ\sim |h - \hc|^{-ψ}$, extracting tunneling exponent $ψ= 0.48 \pm 0.08$ consistent with theoretical predictions ($ψ= 0.5$, $Δχ^2 = 12.3$, $p < 0.001$). This demonstrates that unsupervised learning can identify qualitatively different types of critical behavior, serving as a consistency check on known IRFP physics.

Persistent entropy is a scalar summary of persistence barcodes widely used to detect regime changes, yet there is no account of when a structural change in a barcode must produce a detectable change in entropy. We establish a model-agnostic theorem supplying such conditions. Treating persistence diagrams as random objects indexed by a control parameter, we identify a dispersion-condensation mechanism in the normalized persistence weights and derive an explicit lower bound on the entropy difference between the two regimes, valid with high probability at finite sample size and insensitive to the absolute scale of bar lifetimes. We also give a procedure for verifying the hypotheses on empirical barcodes. Applied to convolutional networks, the criterion shows that the circular organization of learned filters reported by Gabrielsson and Carlsson emerges through a sharp topological phase transition, and locates its onset: within a few hundred iterations on MNIST, but an order of magnitude later on CIFAR-10. The same criterion detects the Kuramoto synchronization and Vicsek order-disorder transitions.

Artificial intelligence and robotic systems are increasingly deployed as interacting collectives of learning agents. This raises the question of whether robotic matter, where many learning agents interact, shows the emergence of collective behaviour. Here we study a paradigmatic model of robotic matter and show the emergence of a range of complex, collective behaviours. Specifically, we study systems composed of stochastic interacting particles, each endowed with a deep neural network that optimises transitions based on its environment. In a one-dimensional system, we show that robotic matter exhibits complex phenomena arising from emergent interactions, including self-organisation into distinct temporal learning regimes, particle species, and long-lived frustrated states with suboptimal reward. We further identify an abrupt, density-dependent change in collective behaviour. Active matter theory suggests that this phenomenon reflects a phase transition with signatures of criticality. Our results establish robotic matter as a platform for novel non-equilibrium physics.

Predicting critical phenomena from limited labeled data remains a challenging task in statistical physics. As percolation theory provides a canonical model for phase transitions with well-established critical exponents, it serves as an ideal benchmark for validating new machine learning frameworks. Here, we introduce a label-efficient learning framework based on a Siamese Neural Network (SNN) to identify phase transitions in three-dimensional site and bond percolation models. Using only 22 labeled probability points drawn entirely from non-critical regions, the method locates percolation thresholds with percent-level accuracy and yields estimates of the critical exponent $ν$ consistent with literature values within statistical uncertainty. Analysis of the learned representations clarifies what the network learns: although trained solely on binary similarity labels, the network autonomously converges to a statistic that coincides quantitatively with the normalized largest-cluster size $S_{max}/L^3$ ($r > 0.99$), the finite-size order parameter of percolation. This underlies the framework's most distinctive capability -- a model trained solely on simple cubic lattices identifies the phase transition in face-centered cubic lattices without retraining. The framework thus offers a complementary route to criticality detection in settings where no quantitative order parameter is explicitly defined or labeled data is scarce.