We introduce a space-time Floquet operator for crystals whose spatial and temporal translations are intertwined. Unlike the conventional Floquet operator, which describes evolution over a full time period, the space-time Floquet operator evolves the system over a fraction of the period paired with a spatial shift that generates the primitive space-time unit cell. Its eigenmode spectrum defines a space-time band structure that unfolds conventional Floquet bands to reveal the consequences of the intertwined crystal symmetry in reciprocal frequency–wavevector space. We relate the topology of these space-time bands to quantized transport phenomena, such as Bloch oscillations and adiabatic charge transport, and uncover an intrinsically fractional charge pump enabled by the primitive space-time cycle. Numerical simulations quantitatively support these predictions with small deviations controlled by the speed of the modulation. We also demonstrate how nonreciprocal parametric resonances are naturally anticipated and accurately predicted by our framework. Our approach applies broadly to both classical and quantum systems with space-time symmetry, including non-Hermitian crystals.
Learning from imperfect data is a central theme in machine learning, connecting practical questions of robustness to fundamental questions of learnability. Here, we examine attribute noise: learning from corrupted inputs while keeping the labels intact, a setting that has received considerably less analytical attention than its label-noise counterpart. We consider two types of corruption models: additive noise and replacement noise. Through experiments with multilayer perceptrons (MLPs) on corrupted classification datasets, we find that neural networks remain robust, maintaining well-above-chance accuracy even when inputs are >90% corrupted—far beyond human recognition. To understand this robustness, we analyze infinite-width networks in the heavy-corruption regime using a mean-field-inspired approach and derive a leading-order decision rule for the classification outcome: The network implements a prototype rule, the nearest-class-mean, assigning each test point to the class whose training-set average it most closely resembles. This leading-order decision rule is universal across a broad range of MLP architectures, holding for any depth, as well as a wide class of activation functions and noise distributions. The same centroid mechanism closely matches finite-width network behavior in our experiments and provides an interpretable and analytically tractable account of why learning can succeed even when individual training examples carry almost no signal.
Angle-resolved photoemission spectroscopy (ARPES) provides the most direct experimental access to electronic band structures and many-body interactions in solids. Beyond energy and momentum, the measured intensity also carries detailed information about the initial-state wavefunction through the complex photoemission matrix element governing the transition to—and scattering character of—the photoelectron final state. Orbital tomography seeks to exploit this sensitivity to reconstruct electronic wavefunctions. However, its quantitative implementation has been hindered by the lack of direct information on the phase of the matrix element and by the nontrivial role of solid-state scattering. Here, we overcome these limitations by achieving an experimental reconstruction of the complex photoemission matrix element in quasi-freestanding monolayer graphene (QFMLG), including both amplitude and phase. By combining circular-dichroism ARPES with linearly polarized measurements within a controlled scattered-wave framework, we determine the full matrix element over a broad kinetic-energy range. The reconstructed amplitudes exhibit pronounced channel-selective resonances and Cooper minima, directly revealing substantial solid-state modifications beyond commonly used simplified models. Leveraging this quantitative reconstruction, we establish QFMLG as an easily available in situ intensity reference for ARPES. Its known complex matrix element enables direct normalization for the product of photon flux and detector response, allowing the determination of photoemission cross sections in other materials under identical experimental geometries. This advances orbital tomography by ARPES to a quantitatively normalized approach that separates the initial-state wavefunction from final-state scattering contributions, and more broadly may provide access to the associated quantum geometry.
