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International Journal of Modern Physics B

Spin and Charge Correlations in the Extended Hubbard-Holstein Model

M. Acquarone, Mario Cuoco, Canio Noce

Published Apr 20, 1999• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217979299001156

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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 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.
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.
We derive an effective polaronic interaction Hamiltonian, exact to second order in perturbation, for the spinless one-dimensional Holstein model. The small parameter is given by the ratio of the hopping term (t)(t)(t) to the polaronic energy $({g}^{2}{\ensuremath{\omega}}_{0})$ in the whole region of validity for our perturbation; however, the exception is the regime of extreme antiadiabaticity $(t∕{\ensuremath{\omega}}_{0}\ensuremath{\leqslant}0.1)$ and small electron-phonon coupling (g<1)(g<1)(g<1) where the small parameter is $t∕{\ensuremath{\omega}}_{0}$. We map our polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model. By studying the mass gap and the power-law exponent of the spin-spin correlation function for our Heisenberg spin model, we analyze the Luttinger liquid to charge-density wave transition at half filling in the effective polaronic Hamiltonian. We calculate the structure factor at all fillings and find that the spin-spin correlation length decreases as one deviates from half filling. We also extend our derivation of polaronic Hamiltonian to ddd dimensions.
The properties of a dilute electron gas, coupled to the lattice degrees of freedom, are studied and compared with the properties of an electron gas at half-filling, where spinless fermions with two orbitals per lattice site are considered. The simplest model which includes both the local electron-lattice interaction of the Jahn-Teller type and the electronic correlations is the E ⊗ β-Jahn-Teller-Hubbard model. We analyze the formation and stability of Jahn-Teller polarons and bipolarons, respectively. Our approach is based on a hopping expansion in the strong-coupling regime. The results are compared with recently published findings for the Hubbard-Holstein model [1, 2]. The special case of the Jahn-Teller-Hubbard model at half-filling is mapped on a spin-1/2 Heisenberg model with phonon-dependent coupling constants. This has been derived within a projection formalism that provides a continued-fraction representation of the Green's function. We study the exact solution for two and three particles and compare it with the effective theory on the infinite lattice with one particle per site. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
"We have studied the extended Hubbard and the Peierls-extended Hubbard models in one dimension using primarily computational methods. We discuss the derivation of the models and the computational methods used to study them, focusing primarily on our extensions over existing numerical methods. In particular, we present the construction of ""operator-loop"" updates for spin-1/2 fermions and the inclusion of electron-phonon interactions in the Stochastic Series Expansion Quantum Monte Carlo method and the use of Quantum Parallel Tempering to improve the quality of the numerical data. For the extended Hubbard model at half-filling, we have confirmed the existence of a recently reported bond-order wave phase for the ground state in a small region of parameter space between the well-known charge- and spin-density wave phases. We have studied in detail the nature of the associated quantum phase transitions between all the phases. Upon doping away from half-filling, the doubly-degenerate, spin and charge gapped long-range-ordered ground states of the model are shown to evolve to a Luther-Emery liquid with no charge gap but a finite spin gap. For the Peierls-extended Hubbard model, which includes both electron-phonon and electron-electron interactions, we have studied the dimerization transition of the half-filled system in the presence of fully quantum mechanical phonons. We show that the transition to a dimerized state occurs only when the electron-phonon coupling exceeds a finite critical value that depends on the bare phonon frequency. This is in contrast to previous studies in similar models with adiabatic phonons that had predicted a dimerized state for arbitrarily small values of the electron-phonon coupling."
We explore the quasiparticle properties of lattice polarons on the basis of a quite general electron-phonon Hamiltonian with a long-range displacement-type interaction. To treat the dynamical quantum phonons without significant loss of accuracy we adapt an exact Lanczos diagonalization method and compute various static and dynamical quantities, such as the electron-lattice correlation function, the polaron band dispersion, the effective polaron mass, the kinetic energy, the single-particle spectral function, and the optical conductivity, on finite one-dimensional lattices for a wide range of model parameters. We compare the results with those obtained for the standard Holstein model with short-range electron-phonon interaction only.