$E=\frac{\hbar^2 k^2}{2m}$) separated from each other.

![A 1D lattice, lattice constant $a$, undergoes a Peierls transition and distorts the lattice such the the lattice constant becomes $2a$. The brown region shows the electronic density.[]{data-label="peierlsfig"}](peierls.pdf)

The point where two dispersions cross each other is a point of degeneracy which will be lifted once weak periodic potential due to the lattice is included. For the similar reason energy gaps arise in their dispersion relation at the Brillouin zone boundary when weak periodic potential is considered. If we put just one electron per lattice site then the states from $-\pi/2a$ to $+\pi/2a$ will be occupied without a band gap between the occupied and the unoccupied states. Hence, the system will still behave like a metal. However, in 1930s, Rudolf Peierls proved that a one-dimensional equally spaced chain with one electron per lattice site is unstable and such a system is prone to distortion. This can be qualitatively understood in the following way.

![Due to the distortion in the lattice, the lattice constant is now $2a$ instead of $a$ therefore the Brillouin zone boundary is at $k=\pm \frac{\pi}{2a}$[]{data-label="pinsulator"}](pinsulator.pdf)

In FIG. \[peierlsfig\], the brown region shows the electronic density due to the electrons. In the 