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