Correlated Quantum Matter Beyond Band Theory: A Continuum-Interaction Formalism for Strongly Coupled Electrons

Abstract

The failure of conventional band theory to describe strongly correlated materials—high-Tc superconductors, Mott insulators, and strange metals—reveals a fundamental incompleteness in our current understanding of electron–electron interactions. In this work, I propose a unified continuum-interaction formalism that treats electronic behavior not as a perturbation around independent quasiparticles, but as an emergent quantum collective governed by non-local correlations. Using an extended Hubbard–Landau functional, a correlation-driven spectral reconstruction model, and a tensor-network-inspired coarse-graining operator, I derive a framework capable of capturing insulating, metallic, and incoherent regimes within a single mathematical structure. This approach suggests that the breakdown of band theory is not an anomaly but an inevitable manifestation of collective entanglement. I discuss analytical consequences, numerical implications, limitations, and future research directions toward a full predictive theory of correlated quantum matter.