= \pm e^{-\frac{1}{2}} e^{-\frac{K}{4\pi}}\end{aligned}$$ It can also be checked, by plugging in $\delta_{\text{min}}$ into $\Delta E_{\text{total}}$, that total change in energy is negative! This means that the system gains energy by distorting itself by $\delta_{\text{min}}$. Note that is was only possible due to electronic energy that compensated the higher-than-original energy contributed by the lattice due to distortion. This is called **Peierls instability**.

As illustrated in FIG. \[pinsulator\], now it is easy to notice that for the distorted lattice, the zone-edges are located at $k = \pm \pi/2a$ instead of at $k = \pm \pi/a$ since now the periodicity of the Brillouin zone is doubled. Doubling the period hence introduces new band gaps in the Brillouin zone and therefore system becomes lower in energy corresponding to the states in the vicinity of the new gaps. This transition makes the band completely filled and a band gap (shown as $2\Delta$ in FIG. \[pinsulator\]) opens up separating the empty states from the filled states at the new edges of the Brillouin zone. This makes the system a band insulator.

Berry Phase {#secbphase}
===========

Let us digress a little and look at a very interesting concept in quantum mechanics that will become necessary at 