Beam-Plasma Collective Oscillations in Intense Charged-Particle Beams: Dielectric Response Theory, Langmuir Wave Dispersion, and Unsupervised Detection via Prometheus

We develop a theoretical and computational framework for beam-plasma collective oscillations in intense charged-particle beams at intermediate energies (10-100 MeV). In Part I, we formulate a kinetic field theory governed by the Vlasov-Poisson system, deriving the Lindhard dielectric function and random phase approximation (RPA) polarization tensor for three beam distribution functions. We prove via the dielectric function epsilon(omega,q)=0 the existence of undamped Langmuir wave modes above a critical beam density n_c, obtain explicit beam-plasma dispersion relations, and show that Landau damping vanishes above the particle-hole continuum. The plasma frequency Omega_p^2 = ne^2/(m*epsilon_0) is fixed by the f-sum rule independently of distribution shape; higher dispersion coefficients depend on velocity moments. Space charge effects drive anomalous beam broadening with sqrt(n-n_c) onset and Friedel oscillations at q=2k_F. The beam-plasma transition belongs to the 3D Ising universality class via renormalization group analysis. In Part II, we validate these predictions using Prometheus, a beta-VAE trained on static structure factor data S(q) from particle-in-cell (PIC) beam simulations. Prometheus detects collective plasma oscillation onset in Gaussian and uniform distributions, confirms their absence in the degenerate Fermi gas (n_c -> 0), and resolves the Kohn anomaly at q=2k_F. Dispersion analysis of S(q,omega) from PIC simulations verifies the distribution-independent Omega_p predicted by the f-sum rule. All six validation checks pass. Predicted signatures -- density-tunable plasma resonances at omega_p proportional to sqrt(n), anomalous beam broadening with sqrt(n-n_c) onset, and Friedel oscillations -- are accessible at existing intermediate-energy beam facilities.

Unsupervised Discovery of Intermediate Phase Order in the Frustrated $J_1$-$J_2$ Heisenberg Model via Prometheus Framework

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 validated on classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the $J_1$-$J_2$ phase diagram via unsupervised analysis of exact diagonalization ground states for a $4 \times 4$ lattice. Through dense parameter scans of $J_2/J_1 \in [0.3, 0.7]$ with step size 0.01 and comprehensive latent space analysis, we investigate the nature of the intermediate regime using unsupervised order parameter discovery and critical point detection via multiple independent methods. This work demonstrates the application of rigorously validated machine learning methods to open questions in frustrated quantum magnetism, where traditional order parameter identification is challenged by competing interactions and limited accessible system sizes.

From Classical to Quantum: Extending Prometheus for Unsupervised Discovery of Phase Transitions in Three Dimensions and Quantum Systems

We extend the Prometheus framework for unsupervised phase transition discovery from 2D classical systems to 3D classical and quantum many-body systems, addressing scalability in higher dimensions and generalization to quantum fluctuations. For the 3D Ising model ($L \leq 32$), the framework detects the critical temperature within 0.01\% of literature values ($T_c/J = 4.511 \pm 0.005$) and extracts critical exponents with $\geq 70\%$ accuracy ($β= 0.328 \pm 0.015$, $γ= 1.24 \pm 0.06$, $ν= 0.632 \pm 0.025$), correctly identifying the 3D Ising universality class via $χ^2$ comparison ($p = 0.72$) without analytical guidance. For quantum systems, we developed quantum-aware VAE (Q-VAE) architectures using complex-valued wavefunctions and fidelity-based loss. Applied to the transverse field Ising model, we achieve 2\% accuracy in quantum critical point detection ($h_c/J = 1.00 \pm 0.02$) and successfully discover ground state magnetization as the order parameter ($r = 0.97$). Notably, for the disordered transverse field Ising model, we detect exotic infinite-randomness criticality characterized by activated dynamical scaling $\ln ξ\sim |h - h_c|^{-ψ}$, extracting a tunneling exponent $ψ= 0.48 \pm 0.08$ consistent with theoretical predictions ($ψ= 0.5$). This demonstrates that unsupervised learning can identify qualitatively different types of critical behavior, not just locate critical points. Our systematic validation across classical thermal transitions ($T = 0$ to $T > 0$) and quantum phase transitions ($T = 0$, varying $h$) establishes that VAE-based discovery generalizes across fundamentally different physical domains, providing robust tools for exploring phase diagrams where analytical solutions are unavailable.