 Solution

A Peierls distortion is a spontaneous lattice modulation that can lower the energy of a one-dimensional metal by opening a gap at its Fermi points. The associated change from metallic to gapped electronic behavior is a Peierls transition. The key wavevector is $2k_F$: it couples states near $+k_F$ and $-k_F$, which otherwise have the same energy.

Consider a spin-degenerate chain with one orbital per atom at spacing $a$ and one Electron per atom. The band is half filled and $k_F=\pi /(2a)$. A distortion of period $2a$ alternates short and long bonds, doubles the primitive cell, and folds the original Fermi points onto the new zone boundary. The distortion can therefore mix the degenerate states precisely where the occupied and empty states meet.

An explicit dimerized tight-binding chain has alternating hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, where $t>0$. In the reduced zone $|k|\le \pi /(2a)$,There are two Electrons per doubled cell, including spin, so the lower band is filled and the upper one empty. Splitting the degenerate states pushes occupied levels downward and empty ones upward, giving an electronic energy gain.

In weak coupling the electronic gain per length has the characteristic form $-C{\delta }^2\ln (t/|\delta |)$, with $C>0$. The elastic cost of moving the ions is proportional to $+K{\delta }^2$. The logarithm makes the undistorted ideal one-dimensional zero-temperature metal unstable to a small distortion even when the electron-lattice coupling is weak. A self-consistent balance selects a finite distortion amplitude. Two opposite dimerization patterns have the same energy for a translation-symmetric isolated chain.

The basic picture links structural modulation to electronic bond order or charge-density modulation, and can open an insulating gap. It does not assert that every real chain has exactly one Electron per atom or period doubling: away from half filling, a modulation at $2k_F$ need not be commensurate with two lattice spacings. Temperature, finite chain length and fluctuations cut off the ideal logarithmic response. Interchain coupling and the real lattice dynamics determine whether a macroscopic finite-temperature ordered phase actually forms. The strictly one-dimensional zero-temperature instability alone is not a proof of finite-temperature long-range order.