# Introduction and previous work

The electronic properties of a material are primarelly dictated by three distinct effects(The Mott Metal-Insula...).

1. The ion potentials,

2. the electron-electron interaction, and

3. externally applied fields

A full solution accounting for all of these interaction would yield a many body problem that would typically be unsolvable. For this reason it is necessary to make approximations to obtain meaningful results. The most common of this approximations would be to ignore electron-electron interactions and assume a stationary periodic potential. This near-free electron problem leads to the Band Structure Theory that is often solved with assistance of Bloch Theorem (Kittel 1966). This solution leads to a very simple classification of metals, and insulators based on their band structure and Fermi Energy.

This method, although often successful, it fails by omitting electron-electron interactions and lattice distortions. A clear example of this is NiO; by its half filed orbitals one would predict this material to be a metal- however it has insulating properties due to electron-electron interactions.

The presence of two opposite mechanism driving that drives to either metallic or insulating behavior can lead to some pretty interesting results. As an example of this consider a N hydrogen atoms distributed at a constant density. The system is described by two non-commuting operators; the Kinetic energy, and the ion potentials. The kinetic energy of the electrons will trend to spread the the electrons, and the ion potential (driven by the coulomb force) will trend to localize the electrons near the protons. The electrons therefore have to phases; bound, or unbound. At low densities, the attractive potential can act over long distances which localizes the electrons. This is an insulating state. As we increase the density we pass a critical density after which the attractive potential is screened by other particles in the systems for the electrons to be bound(Other titles in the {...). This leads to a metallic state at high densities.

This change in properties can be characterized as quantum phase transition. In general a quantum phase transition can be induce by varying the relative strength of two non-commuting operators. It is therefore possible to externally alter the relative strength of these mechanisms by varying an external parameter i.e. pressure (The Mott Metal-Insula...) . This metal-insulator transition as quantum phase transition is shown in figure 1. It is worth noting that proper transition is only attained at zero temperature. This is because at higher temperature there is always some conduction, at which point it becomes a discussion of how good of an insulator it is.

Sketch of a metal-insulator transition as a quantum phase transition for an external parameter $$\lambda$$. Note a proper transition can only be obtained at $$\lambda_c$$ for $$T=0$$. At higher temperature there is only a transition from semiconducting to semimetallic. (from (The Mott Metal-Insula...))

An example of a quantum phase transition would be the Anderson Transition in semiconductors. It stems from the fact that disordered lattice defects can lead to insulation(Anderson 1958). This is better understood as the coherent backscattering from the random defects (Bergmann 1984).This effect is well understood in semiconductors as a sharp increase in conductivity after passing a dopant concentration. Afterwhich there is an increase in conductivity given by a power law(Thomas 1985).

A similar effect can be driven from the competition between internal energy and the entropy of a system such that an ordered system is formed at low temperatures. This breaks the continuous symmetry of the Hamiltonian and changes the electronic properties. We can refer to this type of transition as a thermodynamic phase transition, since it is thermodynamically driven.