This is however not the case when electron-electron interactions cause the electrons to localize on to individual ions of an ionic insulator. In such cases, the traveling wave picture of electronic states is not valid, and a much better description is a local one whereby one imagines a lattice of ions, each with an optimal charge on it. Movement of electrons is forbidden by the fact that such motion involves transferring charge from one ion to another and is energetically very "expensive'' due to the large charging energy of these little "ionic capacitors''. Such insulators are called "Mott insulators'' after Sir Neville Mott who first described the mechanism that forces the electron fluid to make a transition to an insulating state.

In such Mott insulators, the charge of an electron is no longer an active participant in the low temperature physics since it is "frozen out'' by the requirement that each ion have an optimal energetically favourable charge configuration that minimizes the Coulomb repulsion energy between electrons. However, in many such Mott insulators, the optimal number of electrons on some ions can be odd. In such cases, the ion has a spin degree of freedom (we are imagining that the orbital angular quantum number is quenched by crystal field effects familiar from solid state physics). These spins are typically coupled antiferromagnetically to each other (we will see exactly how in more detail later), and the low energy physics is thus determined by the behaviour of a system of interacting quantum mechanical spins.

Thus, the principal players in our story are itinerant fermions and bosons, and quantum mechanical spins which may be thought of as electrons that have lost the ability to hop from ion to ion. Starting with the next lecture we will introduce a very convenient path-integral formalism for working with the partition function for bosons, fermions or spins, and computing various properties of such systems in equilibrium. Then we will take up an example and examine in more detail the issues of symmetry breaking and stability associated with phases that develop long-range order when the equilibrium state breaks some global symmetry of the underlying Hamiltonian.