Global Fractional Quantization in Condensed Matter Systems: A Unified Framework for Collective Quantum Phenomena

This paper presents a systematic extension of the Global Fractional Quantization (GFQ) framework from particle physics to condensed matter systems. We establish a rigorous theoretical foundation for the quantization of collective observables in many-body quantum systems, demonstrating that sums of specific physical quantities over entire bands, ensembles, or collective states exhibit discrete, rational relationships. The framework is built upon four fundamental postulates—Gauge Constraint, Gauge Invariance, Gauge Periodicity, and Global State Symmetry—applied to appropriately defined U(N) collective multiplets in condensed matter contexts. We provide detailed mathematical derivations with complete proofs and demonstrate the framework’s applicability to several key condensed matter systems: topological insulators, superconductors, magnetic systems,and correlated electron materials. The theory predicts novel quantization conditions for sums of Berry curvatures, total polarization, collective spin moments, and superconducting phase differences,which we validate through both analytical models and numerical calculations with comprehensive error analysis. Our results reveal a hierarchical structure of quantization laws that correlate with the system’s dimensionality, symmetry, and fundamental nature (topological vs. trivial, gapped vs. gapless). The GFQ framework offers a new unifying perspective on emergent quantization phenomena across diverse condensed matter platforms and provides testable predictions for future experiments.