Quantum Geometry, Fractionalization, and Provability Hierarchy: A Unified Framework for Strongly Correlated Systems

TL;DR

This paper introduces a unified framework for strongly correlated systems, revealing geometric, topological, and fractionalized phenomena, with predictions verified by numerical methods and implications for quantum material classification.

Contribution

It presents five novel discoveries linking quantum geometry, fractionalization, and computational complexity in strongly correlated systems.

Findings

Prediction of golden-ratio scaling of quantum metric fluctuations near Mott criticality

Fibonacci sequence of allowed fractional Chern insulator charges

Classification of critical states as 'true but unprovable' QMA-hard problems

Abstract

Mott physics - the interplay between itinerancy and localization of electrons - is undergoing a paradigm shift from the binary "bandwidth - filling" tuning framework to an intertwining of geometric, topological, and fractionalized degrees of freedom. Based on a series of breakthroughs in 2024 - 2025, this paper proposes five pioneering discoveries: (1) Prediction of the golden-ratio scaling of quantum metric fluctuations near the Mott critical point, supported by functional renormalization group arguments and DMRG numerical verification (phi = 0.618 +/- 0.005); (2) Establishment of a correspondence between the denominator q of fractional Chern insulator charge and the subgroup index of the quantum geometry group, predicting that allowed q values follow the Fibonacci sequence {2,3,5,8,13,...} with specific material realizations; (3) Proposal of the Provability Hierarchy Theorem, classifying critical states like strange metals as "true but unprovable" QMA hard problems, establishing a rigorous connection to the complexity of the Consistency of Local Density Matrices(CLDM) problem; (4) Prediction of interference oscillations in the nonlinear Hall conductance within the pseudo gap phase, induced by geometric phase differences, supported by tight-binding numerical simulations; (5) Unveiling the quantum geometric tensor as a unified descriptor of band geometry and topology. These findings provide an experimentally testable theoretical framework for understanding strongly correlated quantum materials.