Peierls-Hubbard model

Group method was applied for dynamic aL and yL using ID Hubbard and U-V models and a Hubbard-Peierls model Hamiltonian in the case of linear polyenes. 

The symmetrized Density Matrix Renormalisation Group Procedure145 using a Hubbard-Peierls model Hamiltonian has given for a linear polyene chain quite satisfactory results for aL(static) and for yL(coa co, co2, ) (coa = co + a>2 + ) and for (yL/jV)(static). 

Pang and Liang have calculated the charge excitation gap for the Hubbard-Peierls model, which is defined for a neutral chain of N sites (number of electrons Ng = N) as Eg N) E Ne = N + l)+E Nc = N — 1) — 2E Ne = N) [105]. We note that this is not the optical gap in the extended Hubbard-Peierls model, since the definition excludes the contribution due to Coulomb attraction between electron and hole. Instead, this quantity becomes our definition of charge excitation band gap which constitutes the continuum edge, the reason being, taht at the band edge, the electron and hole are uncorrelated... 

For P = 0, as expected for the Hubbard-Peierls model, the binding energy Ef, is always calculated to be a small negative quantity 0 1/N) for any U and 6. This excellent agreement with the physical picture serves as a check on our numerical scheme. 

The Hubbard model, which has played a significant part in the earlier development of nonlinear optical response theory of molecules, has been revived by Shehadi et al.203 to explain the properties of small bridged metallic polymers. The Hubbard-Peierls hamiltonian has also been used by Shuai et al.,204 in conjunction with a symmetry adapted density matrix renormalization group formulation, to calculate a number of properties, including third harmonic generation in trans-octatetraene. 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

This note is organized as follows. In section II, the RVA is presented using the example of the two-legspin ladders. In section III, the RVA method is applied to the study of the dimerized chain described by the Extended Peierls-Hubbard model. A nice similarity between the two-leg ladder and this system is pointed out and comparisons with DMRG calculations are made. We conclude in section IV and give some reasonable perspectives. 

As mentioned in Section II, LRO in two dimensions can exist only for a real order parameter, that is, for CDW in a half-filled band. This would be the case for BOW in the polymers or the Peierls state, which would be stabilized by transverse hopping or interchain coupling. This is also the case of the CDW state of the n = 1 two-dimensional Hubbard model. All other types of instabilities, such as those treated in the RPA previously in Section V, require three-dimensional coupling to stabilize any LRO. 

Let us now introduce the electron-vibration interaction in the symmetric dimer, to account for the modulation of the site energies, and of the electron transfer integral, by the intramolecular vibrational modes and by the rigid molecular motions respectively. This leads us to the so-called Peierls—Hubbard model. ... 

For a non-regular chain, the situation becomes even more complex, because of the possible superposition of two different energy gaps, resulting, respectively, from a splitting in the energy scale, and from a reduction of the reciprocal lattice these are the so-called Peierls-Hubbard models. 

Conjngated polymers differ from crystalline semiconductors and metals in several aspects and are often treated theoretically as a one-dimensional system. The formation of the band gap is explained taking into account either electron-phonon interactions or electron-electron interactions among 7t-electrons. If electron-phonon interaction dominates in real 7t-conjugated polymers, these systems could be treated using Peierls theory. In contrast, when electron-electron interactions dominate, the Hubbard model could be used to explain the physical properties of polymers. 

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