Undimerized chains

In an undimerized chain a spin-density wave exhibits gapless spin excitations and gapped charged excitations. However, in a dimerized chain all three types of order exhibit gapped spin and charge excitations. In fact, for a dimerized chain the spin-density and bond order waves coexist. Mazumdar and Campbell have shown (Mazumdar and Campbell 1985) that the Pariser-Pople-Parr model will exhibit a broken-symmetry ground state provided that. 

Luttinger liquid behaviour applies to metallic systems. However, as already discussed, for a half-filled band the metallic state is unstable with respect to a broken symmetry spin-density wave ground state. There is a gap to charge 

is the retarded single-particle Green function, 

Since the charge gap is the gap between the highest electron removal state and the lowest electron addition state, we can also define it as 

Hubbard band. We describe excitons in detail in the next chapter, although a brief description of the energy of bound states in the weak-couphng, undimer-ized limit is given in Section 5.2.2. For polymers with inversion symmetry the lowest charge-transfer (or ionic) excitation is the state. Another kind of 

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In the context of fra/u-polyacetylene cjia and c are, respectively, the creation and annihilation operators of an electron with spin projection a in the n-orbital of the nth carbon atom (n= l,...,N) that is perpendicular to the chain plane (see Fig. 3-3). Furthermore, u is the displacement along the chain of the nth CH unit from its position in the undimerized chain, P denotes the momentum of this unit, and M is its mass. 

The Coulomb interaction between the 7r-electrons is neglected. The standard trani -polyacetylene parameters are to = 2.5eV for the hopping amplitude in the undimerized chain, a=4.1 eV/A for the electron-phonon coupling, and A =21 eV/A for the spring constant [1, 4, 8]. 

Fig. 5. Electronic energy vs. dimerisation coordinate u in trans-poly acetylene the two equal minima correspond to the A and B structures. The top of the barrier C would correspond to the undimerized chain C.

The band structure is shown in Fig. 3.4. e = 2t at fc = 0 and e = 2 t at A = 7r/2a. Thus, the band gap is 45t, while the full band width is the same as the undimerized chain, namely 4t. Notice that as 6 is increased from 0 to 1 the band structure evolves from that of the undimerized chain (with a folded dispersion), to localized orbitals on the double bonds, with energies of 2t. 

The energies of the valence and conduction bands for open, dimerized chains are again given by eqns (3.23) and (3.24), but with k replaced by j3 (Leimard-Jones 1937). However, now, unlike the case for undimerized chains, there is no closed expression for [3. Instead, (3 is determined by the transendental equation. 

In the intermediate-coupling regime Mott-Wannier excitons are the more appropriate description for large dimerization (J = 0.2), while for the undimerized chain Mott-Hubbard excitons are the correct description. For dimerizations relevant to polyacetylene and polydiacetylene (that is, S ... 

The effects of electron-phonon interactions alone were described in Chapter 4. We showed that these interactions lead to a dimerized, semiconducting ground state and to solitonic structures in the excited states. On the other hand, the effects of electron-electron interactions in a polymer with a fixed geometry were described in Chapters 5 and 6. There it was shown that the electronic interactions cause a metal-insulator (or Mott-Hubbard) transition in undimerized chains. Electron-electron interactions also cause Mott-Wannier excitons in the weak-coupling limit of dimerized chains, and to both Mott-Hubbard excitons and spin density wave excitations in the strong coupling limit. 

As described in Section 3.3.1, for cyclic undimerized chains this Hamiltonian is diagonalized by the Bloch states,... 

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