Peierls theorem

This splitting of the band is an example of Peierls theorem, which asserts that a onedimensional metal is always electronically unstable with respect to a... 

Infinite linear polyenes show a bond alternation between successive long and short C-C bonds [1], a consequence of the Peierls theorem on the nonexistence of one-dimensional metals [2], This Peierls distortion (or instability) is very important both from a theoretical and a practical point of view, being a typical example of a metal-insulator transition [3]. Consider an infinite chain of equally spaced sites -(CH)-, each of them bearing one electron in a single valence orbital. In this case we have a half-filled band and the system has metallic character. If we distort the chain into an alternating sequence of short and long bonds -(CH=CH)-, the half-filled band splits into a lower one completely filled and an upper empty band, separated by a gap. This dimerized polyacetylene is an insulator. 

One-dimensional chains of equal objects are thermodynamically unstable to a static or dynamic distortion. In 1955 Peierls introduced the theorem [70], now named after him, which states that such a chain will distort spontaneously and dynamically to minimize the energy the distortion will become static (i.e. permanent), rather than dynamic, if there is enough "off-chain coupling" with the lattice phonons. This Peierls theorem can be thought of as Ae... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

Electron-phonon coupling plays a crucial role in one-dimensional systems. For any value of the electron-phonon coupling an infinite, undistorted polymer chain is unstable with respect to a lower symmetry, distorted structure. This is a consequence of the well-known Peierls theorem (Frohlich 1954 Peierls 1955), which states that a one-dimensional metal is unstable with respect to a lattice distortion that opens a band gap at the Fermi surface. A proof of bond-alternation in conjugated polymers in the noninteracting limit was first presented independently by Ooshika (1957, 1959), and Longuet-Higgins and Salem (1959). 

Bond alternation in polyenes is an example of a more general theorem for one-dimensional crystals called Peierls theorem. This theorem applies to systems like polyenes, where there is one orbital and one electron per atom, that is, a half-filled band. The theorem was first stated in 1955 by Rudolf Peierls. In the case of one-dimensional systems and half-filled bands, we use a proof by Lionel Salem. The Hamiltonian is expanded in a Taylor series for a geometry with equal bond lengths ... 

Physicists call this appearance of short bonds and long bonds dimerization, a notation that has very little to do with dimerization in chemistry. However, Peierls theorem can also be applied to a stack of organic tt-systems, for example, TCNQ. In this case, it is useful to talk about a multimer that forms dimers. 

Fig. 7-1 shows one of the ways of illustrating the formation of this Peierls gap. While the Peierls theorem has been found to hold well for experimental systems in general, a caveat is to be noted It correctly predicts the formation of a bandgap only when no other mechanism exists to create one. (Among possible alternative mechanisms are electron-phonon interaction to create a superconducting gap, or... 

Now as regards the statements made by theory, Peierls requires that [L] should decrease towards the absolute zero, as we explained above (section 9, p. 57). But Peierls also holds that the constancy of Z at low temperature is intelligible, if the additivity of the ideal resistance and the residual resistance . .. is strictly valid, both for electrical conductivity and thermal conductivity For the classical theorem on the number of collisions is justified even at low temperatures for the statical lattice disturbances regarded as causing the resistance, whereas it is not so for the thermal agitation here. 

We seek a solution of H for arbitrary A . In Section 4.4 we discuss the Hellmann-Feynman theorem, which gives us a general solution for any eigenstate. For now, however, we describe the Peierls mechanism, which gives us the dimerized, broken-symmetry ground state. 

Regular polyacetylene is a one-dimensional conductor, in that there is no band gap between the occupied and virtual sets of MOs. This type of system is unstable, and, according to Peierl s theorem, the electronic structure should distort so that a charge density wave is formed. 

---