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

shifa liu

Published Dec 22, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.33774/coe-2025-v1hsf

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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.
The principle of symmetry is a cornerstone of modern physics, yet its role in condensed matter many-body systems has largely been qualitative, serving as a classificatory tool. In this work, we establish, for the first time, a universal framework of hierarchical fractional quantization (HFQ) for superconducting and superfluid systems with continuous symmetries and quantum fluctuations.We demonstrate that symmetry can impose rigorous, computable quantitative constraints on the numerical values of physical observables.We apply this framework to predict the response of the d-wave order parameter in high-temperature superconductors under uniaxial stress (predicting R = 1 and deriving new constraints on superfluid density anisotropy) and the excitation spectrum splitting of a Bose superfluid in an optical lattice under lattice distortion (predicting R = 1/2). Systematic quantitative validation using multiple published experimental datasets shows excellent agreement with theoretical predictions. This work elevates symmetry from a qualitative organizing principle to a quantitative predictive tool, offering a new paradigm for understanding competing orders and symmetry breaking in strongly correlated quantum materials.
The principle of symmetry serves as a cornerstone of modern physics, yet its role in condensed matter many-body systems has largely remained qualitative, acting primarily as a classificatory tool. In this work, we establish, for the first time, a universal framework of hierarchical fractional quantization (HFQ) for superconducting and superfluid systems with continuous symmetries and quantum fluctuations. We demonstrate that symmetry can impose rigorous, computable quantitative constraints on the numerical values of physical observables.We apply this framework to predict the response of the d-wave order parameter in high-temperature superconductors under uniaxial stress (predicting R = 1 and deriving new constraints on superfluid density anisotropy) and the excitation spectrum splitting of a Bose superfluid in an optical lattice under lattice distortion (predicting R = 1/2). Systematic quantitative validation using multiple published experimental datasets shows excellent agreement with theoretical predictions. This work elevates symmetry from a qualitative organizing principle to a quantitative predictive tool, offering a new paradigm for understanding competing orders and symmetry breaking in strongly correlated quantum materials.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
This paper presents a novel geometric quantum field theory (GQFT) framework for analyzing quantum field theories within condensed matter systems. We introduce a new approach that explicitly incorporates geometric constraints and topological properties, aiming to provide a deeper understanding of emergent phenomena observed in these systems. The core of this work involves extending standard field theory techniques to accommodate these geometric elements, thereby offering a fundamentally different perspective on the behavior of quantum fields within condensed matter. We detail the key mathematical components and demonstrate the framework's ability to successfully model a range of complex quantum phenomena, including the emergence of quantized vortices and topological phase transitions. The resulting theory provides a robust and mathematically sound tool for investigating the intricate interplay between quantum mechanics and geometric structures in condensed matter physics. This research will significantly advance the field by offering a more intuitive and powerful method for tackling problems previously intractable with traditional approaches.
We develop a conditionally closed theoretical architecture for condensed-matter and solid-state physics from three physical axioms: microscopic ontic definiteness and finite localization, the physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. Electrons, nuclei, atoms, and molecular units enter as finite topological carriers, whereas crystals, energy bands, phonons, quasiparticles, order parameters, defects, and topological phases are derived as spectral, variational, statistical, or homotopy properties of the resulting many-body system. We construct a sorted first-order language, a model-theoretic conservative extension of the underlying physical theory, and a constrained Legendre descent from a unified matter–substrate action to a self-adjoint, lower-bounded many-body Hamiltonian. A signed ownership ledger prevents double counting among self-energy, screening, correlation, electron–phonon, electromagnetic, historical-memory, and environmental contributions. For neutral stable branches, we establish a thermodynamic limit under translation covariance, two-sided almost-additivity, controlled Coulomb surface terms, and passive auxiliary realizations of causal memory. The framework derives crystal symmetry, Bloch and Wannier representations, lattice dynamics, phonon quantization, electronic bands, semiconductor statistics, Green functions, conserving Kadanoff–Baym dynamics, strong-correlation reductions, magnetism, polarons, superconductivity, disorder, localization, transport, critical phenomena, dielectric response, defects, interfaces, and low-dimensional systems. Conventional Bloch, tight-binding, nearly-free-electron, Debye, Landau, Fermi-liquid, Hubbard, Heisenberg, BCS/BdG, Boltzmann, and Kubo theories are recovered only as controlled limits with explicit validity domains and failure conditions. Fundamental-carrier topology, band topology, and interacting many-body topology are distinguished rigorously. Stability criteria are given for spectral and mobility gaps, Green-function invariants, Floquet and non-Hermitian phases, completely positive open dynamics, Liouvillian steady-state bundles, and finite-memory auxiliary systems. We also specify when thermodynamic, zero-frequency, zero-wave-number, long-time, symmetry-breaking, and regulator limits commute and when their ordering defines distinct physical observables. A terminal completion theorem identifies the conditions for pure-theory closure: semantic conservativity, action traceability, self-adjoint lower-bounded dynamics, thermodynamic-limit control, causality, positivity, conservation laws, controlled standard limits, valid topological domains, and falsifiable observable maps. This theoretical closure does not imply that the spectrum, phase diagram, or transport coefficients of any particular material have been calculated or experimentally confirmed; such tasks belong to separate material-application programs. Keywords **global realism; topological condensed matter; solid-state physics; many-body theory; causal memory; historical disturbance field; Green functions; strong correlations; superconductivity; topological phases; quantum transport; thermodynamic limit; open quantum systems; Liouvillian topology; controlled effective theories**
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