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arXiv:cond-mat/0010465 · Physical review. B, Condensed matter

Variational study of the extended Hubbard-Holstein model on clusters of variable site spacing

M. Acquarone, Mario Cuoco, Canio Noce, Alfonso Romano

Published Jan 2, 2001• 6 citations
arXivHTMLPDF

Abstract

We study the complete extended Hubbard-Holstein Hamiltonian on a four-site chain with equally spaced sites, with spacing-dependent electronic interaction parameters evaluated in terms of Wannier functions built from Gaussian atomic orbitals. By successive application of generalized displacement and squeezing transformations, an effective polaronic Hamiltonian is obtained, containing a purely phonon-induced, long-range intersite charge interaction. The phase diagrams for $N=1,$ 2, 3, and 4 electrons are determined by variational minimization of the sum of the electronic and the phononic energy, paying special attention to the effects of the above-mentioned phonon-induced long-range interaction. To characterize the physics of each ground state, we evaluate how variations of the site spacing affect the behavior of several correlation functions of experimental interest, with the aim of representing lattice deformation effects, either spontaneous or induced by external pressure.

DOI
10.1103/physrevb.63.035110

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We study the complete extended Hubbard-Holstein Hamiltonian on a four-site chain with equally spaced sites, with spacing-dependent electronic interaction parameters evaluated in terms of Wannier functions built from Gaussian atomic orbitals. By successive application of generalized displacement and squeezing transformations, an effective polaronic Hamiltonian is obtained, containing a purely phonon-induced, long-range intersite charge interaction. The phase diagrams for N=1,N=1,N=1, 2, 3, and 4 electrons are determined by variational minimization of the sum of the electronic and the phononic energy, paying special attention to the effects of the above-mentioned phonon-induced long-range interaction. To characterize the physics of each ground state, we evaluate how variations of the site spacing affect the behavior of several correlation functions of experimental interest, with the aim of representing lattice deformation effects, either spontaneous or induced by external pressure.
From the complete extended Hubbard-Holstein Hamiltonian we obtain a polaronic effective Hamiltonian by successive application of generalized displacement and squeezing transformations, with wavevector dependent characteristic parameters which make possible a phonon-induced long range interaction between charges, whose sign can change with the distance. As an application we consider the four-site chain, in which the bare electronic interactions are evaluated by model Wannier functions of gaussian shape. The ground states for different ranges of interactions are obtained by exact diagonalization of the effective Hamiltonian, followed by simultaneous and independent optimization of the displacement and squeezing parameters. We consider the behaviour of the static spin and charge correlation functions for the half-filled case, upon varying the shape of the Wannier functions for given electron-phonon coupling. Special attention is devoted to the influence of the phonon coupling on the presence of competing or coexisting charge- and spin-ordered states.
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.
In this paper we have applied the correlated squeezed-state approach to investigate the ground-state properties of the one-dimensional Hubbard model with a local Holstein-type electron-phonon interaction. We have introduced a variational ground state in which the phonon subsystem is in a correlated squeezed state. The correlated squeezed state has built in the correlation between different phonon modes induced by the linear coupling with the electrons. With optimal values of the variational parameters the correlated squeezed state will by construction yield an energy lower than those obtained in earlier treatments. This means that our variational ansatz is more stable as the ground state of the system. Furthermore, our variational study shows that in the correlated squeezed state the polaronic reduction effect of phonons is much more alleviated, and thus the mass enhancement inherent to the polaron effect is considerably weakened. This weakening of the reduction effect should, in turn, affect other physical properties of the system.
We have explored the properties of a two-fermion system interacting with the phonon field in the framework of the one-dimensional discrete Holstein-Hubbard model. The variational method employed here introduces the correlations between the phonons and the electron wave functions, therefore the composite trial state is not factorized. As a consequence our approach gives reliable results in the whole space of the parameters. We will direct our attention to a new class of solutions found in the intermediate range of values of the nonadiabaticity parameter \ensuremath{\gamma}. These solutions are characterized by anomalous (non-Gaussian) fluctuations of the position of the oscillators. An intersite bipolaron with a relatively small effective mass is stable in a wide region of the parameters due to both exchange and nonadiabaticity effects.
A simple variational displacement phonon basis, obtained through the modified Lang-Firsov (MLF) transformation, is proposed to study the Holstein model.This phonon basis contains only one variational parameter, but capable of describing lattice distortions at distant sites from the charge carrier. Perturbation method based on this MLF basis is employed to calculate the single-electron ground-state energy and static charge-lattice deformation correlation as a function of electron-phonon coupling. The energy obtained up to the second-order perturbation within this approach agrees well with the available numerical results for the entire range of coupling strength.
The polaron features for long-range electron–phonon interaction are investigated by extending a variational approach previously proposed for the study of systems with local coupling. The ground state spectral weight, the average kinetic energy, the mean number of phonons, and the electron–lattice correlation function are discussed for a wide range of model parameters focusing on the adiabatic regime and comparing the results with the short-range case (Holstein model). A strong mixing of electronic and phononic degrees of freedom for small values of the electron–phonon coupling constant is found in the adiabatic case due to the long-range interaction. Finally a polaron ‘phase diagram’ is proposed.