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arXiv:cond-mat/0010164 · International Journal of Modern Physics B

A NON-PERTURBATIVE TREATMENT OF THE GENERALIZED SU-SCHRIEFFER-HEEGER HAMILTONIAN ON A DIMER

Matteo Cococcioni, M. Acquarone

Published Oct 30, 2000• 2 citations
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Abstract

Starting from the Hamiltonian for a dimer which includes all the electronic and electron-phonon terms consistent with a non-degenerate orbital, by a sequence of displacement and squeezing transformation we obtain an effective polaronic Hamiltonian. The renormalized electronic interactions differ from the results of semiclassical or perturbative treatments. The properties of the variationally determined ground state of two particles in the orbital are discussed for variable dimer length in the adiabatic limit

DOI
10.1142/s0217979200003162

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Journal of Physics Condensed Matter•Mar 22, 2001•11 citations
Starting from the Hamiltonian for a dimer which includes all the electronic and electron-phonon terms consistent with a non-degenerate orbital, by a sequence of displacement and squeezing transformation we obtain an effective polaronic Hamiltonian. The renormalized electronic interactions differ from the results of semiclassical or perturbative treatments. The properties of the variationally determined ground state of two particles in the orbital are discussed for variable dimer length in the adiabatic limit
We consider a model Hamiltonian for a dimer of length aaa including all the electronic one- and two-body terms consistent with a single orbital per site, a free Einstein phonon term for a frequency $\ensuremath{\Omega},$ and an electron-phonon coupling g0{g}_{0}g0​ of the Holstein type. The bare electronic interaction parameters were evaluated in terms of Wannier functions built from Gaussian atomic orbitals. An effective polaronic Hamiltonian was obtained by an unrestricted displaced-oscillator transformation, followed by evaluation of the phononic terms over a squeezed-phonon variational wave function. For the cases of quarter-filled and half-filled orbitals, and over a range of dimer length values, the ground state for given g0{g}_{0}g0​ and $\ensuremath{\Omega}$ was identified by simultaneously and independently optimizing the orbital shape, the phonon displacement, and the squeezing effect strength. As aaa varies, we generally find discontinuous changes of both electronic and phononic states, accompanied by an appreciable renormalization of the effective electronic interactions across the transitions, due to the equilibrium shape of the wave functions strongly depending on the phononic regime and on the type of ground state.
For a dimer with a non-degenerate orbital built from atomic wave functions of Gaussian shape we evaluate all the electron-phonon couplings derived from the one-body and two-body electronic interactions, considering both the adiabatic and extreme non-adiabatic limit. Not only the values of the coupling parameters in the two limits, but also the expressions of the corresponding terms in the Hamiltonian differ. Depending on the distance between the dimer ions, some of the two-body couplings are comparable, or even larger than the one-body ones.
We consider a polaron localized on a trap in a one-dimensional lattice with a harmonic potential of nearest-neighbor interaction. We study the lattice oscillations in the framework of quantum mechanics and regard the polaron energy as a functional of the wave function. We take the electron—phonon interaction into account in the framework of the linear Su—Schrieffer—Heeger approximation. We show that the results do not differ from the adiabatic consideration if the lattice oscillations are described classically. For the polaron ground state, we obtain approximate analytic expressions that agree well with the results of numerical simulation.
We study the nature of the ground state of the Su-Schrieffer-Heeger model for electron-phonon interactions in one dimension in the half-filled-band case. We consider the cases of spinless electrons (n=1n=1n=1) and spin-\textonehalf{} electrons (n=2n=2n=2), and discuss the stability of the Peierls-dimerized ground state as a function of the ionic mass and electron-phonon coupling constant. We first consider the zero-mass limit of the theory and extend our results to finite mass using renormalization-group arguments. For spinless electrons, it is found that quantum fluctuations destroy the long-range dimerization order for the small electron-phonon coupling constant if the ionic mass is finite. For spin-\textonehalf{} electrons, the system is dimerized for an arbitrary coupling constant and phonon frequency. Renormalization-group trajectories show that the low-energy behavior of the system is governed by the zero-mass limit of the theory, an nnn-component Gross-Neveu model. Monte Carlo simulations are performed for the model at finite phonon frequencies, and the results are compared with the static limit result. We study in particular the set of parameters appropriate for polyacetylene and find a 15% reduction in the phonon order parameter due to fluctuations of the phonon field. A finite-size scaling analysis of the numerical data for the cases n=1and2n=1 \mathrm{and} 2n=1and2 is performed, which confirms the results obtained from the renormalization-group analysis.
The ab initio description of evolutionary processes in extended electron-phonon systems (polaronic transport, excitonic transfer, etc) up to the present is beyond numerical accessibility, since it requires the simultaneous knowledge of all eigenfunctions and eigenvalues. Therefore, usually rough approximations are made, such as a semiclassical treatment. However, as we have shown in a recent paper, the full quantum-mechanical treatment drastically deviates from the semiclassical approximation (even in a qualitative manner). In the concept discussed here unitary product transformations are introduced, the constituents of which account for the two antagonistic tendencies inherent in every coupled electron-phonon Hamiltonian. We apply our procedure to the concrete case of the dimer-oscillator model by choosing for each of the antagonistic tendencies respectively a one parameter unitary operator, such that full analytical diagonalization is reached in the opposing limits of the Hamiltonian constituents. In the intermediate regime the two parameters of the transformation are suitably optimized. In this manner the generation of the full spectrum of eigensolutions involves two analytically fixed parameters only. The evolutionary behaviour resulting from our procedure is contrasted with the exact numerical result as well as with the one from the semiclassical approach and also with a more simple (`displacive') unitary transformation frequently used in the literature. It is shown that our calculation approaches the exact result in a satisfactory manner in all intrinsic physical parameter regimes (coupling and transfer) and overcomes the drastic shortcomings of previous calculations.