\pm \pi/a$ are called the Brillouin zone edges which have interesting features when weak interactions are introduced - we will come back to this in the next section. If we stick to non-interacting particle description and assume every atom has a single loosely bound electron that hops around, then the band formed by the dispersion relation is half filled (red region in FIG. \[tightbindingdispersion\]). This is because every atomic state contributes one electron whereas its maximum occupancy is restricted to two (due to Pauli’s principle). Notice that in condensed matter, energies are always measured from the Fermi energy so it is convenient to set Fermi energy to zero. Since there are states available at arbitrary small energies above Fermi energy, this system thus describes a metal otherwise, an insulator as we will see in the next section.

Peierls instability {#sec:peierls}
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In the previous problem, we studied a one-dimensional non interacting tight-binding model that we saw described a metal. When we have drawn the $E$ vs. $k$ relation in FIG. \[tightbindingdispersion\], we assumed the existence of background lattice which give rise to a periodic potential that the electrons can sense. Now consider several free electron dispersions (which is just parabolic 