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Mathematical descriptions of the interplay between strong electroncorrelations and lattice degrees of freedom are of enormous importance in thedevelopment of new devices based on metal oxides such as VO$_{2}$ and theCuprate superconductors. In this work the physics of tight-binding typeelectron momentum states interacting with lattice fluctuations is reformulatedinto an approach based on lattice QCD. Strong electron correlations act as asource for phonons, which are incorporated by using SU(2) bosons acting onneighbouring atomic sites. This allows the system to be described by aHamiltonian which describes strong interactions between SU(2) Yang-Mills bosonsnear T$_{c}$ resulting from electron correlations. Monte Carlo and GWcalculations show that at low Temperature the electron-electron interactionsdrive the system into a phase coherent phonon state, breaking the latticesymmetry, and a band gap opens. This formalism is intrinsically able to combinestrong-electron correlations with lattice fluctuations in a manner whichdescribes symmetry-breaking structural phase transitions which manifest spin-and charge ordering.

A Wilsonian Approach to Crystal Structure Transformations driven by Strong Electron Correlations