Modified Lang Firsov Research Articles

Exact solutions of the Holstein model with different site energies

The two-site Holstein model with different site energies, the simplest model to mimic disorder, is analytically studied by a diagonalization method of coherent states. The solutions obtained in our work are exact, and agree well with those derived by a modified Lang-Firsov (MLF) method in the weak- and strong-coupling regimes. The deviation from the MLF solution in the intermediate-coupling regime implies that our solution, including the higher order correlation terms overlooked in the MLF treatment, gives a more accurate description. To check the validity of our approach, we have also calculated the case with the same site energies, which shows that our exact solution could be consistent with a previous treatment of the Holstein model using the coherent expansion method [Phys. Rev. B 65, 174303 (2002)]. The method and the results in the present work would be useful for testing various approximate methods for the Holstein model, and are applicable to more complicated situations.

Effect of different site energies on polaronic properties

Using the perturbation method based on a variational phonon basis obtained by the modified Lang-Firsov (MLF) transformation, the two-site single polaron Holstein model is studied in presence of a difference in bare site energies ($\epsilon_d$=$\epsilon_2$-$\epsilon_1$). The polaronic ground-state wave function is calculated up to the fifth order of perturbation. The effect of $\epsilon_d$ (acts as a site-energy disorder) on the polaron crossover, polaronic kinetic energy, oscillator wavefuncion and polaron localization are studied. Considering a double-exchange Holstein model with finite $\epsilon_d$, role of disorder on the properties of the double-exchange system is also discussed.

Polaron crossover in molecular solids

An analytical variational method is applied to the molecular Holstein Hamiltonianin which the dispersive features of the dimension dependent phonon spectrumare taken into account by a force constant approach. The crossover between alarge and a small size polaron is monitored, in one, two and three dimensionsand for different values of the adiabatic parameter, through the behaviour of theeffective mass as a function of the electron–phonon coupling. By increasing thestrength of the intermolecular forces the crossover becomes smoother and occurs athigher e–ph couplings. These effects are more evident in three dimensions. Weshow that our modified Lang–Firsov method starts to capture the occurrence of apolaron self-trapping transition when the electron energies become of order of thephonon energies. The self-trapping event persists in the fully adiabatic regime. Atthe crossover we estimate polaron effective masses of order times the bare band mass according to the dimensionality and the value of the adiabaticparameter. Modified Lang–Firsov polaron masses are substantially reduced intwo and three dimensions. There is no self-trapping in the antiadiabatic regime.

Nature of the self-trapping transition in a one-dimensional Holstein–Hubbard model

The nature of the self-trapping transition in a many-polaron system is investigated within the framework of an extended Holstein–Hubbard model in one dimension using a variational method. The phonon degrees of freedom are first eliminated employing a modified Lang–Firsov transformation and an onsite squeezing transformation and then using a correlated squeezed phonon state to yield an effective Hubbard model which is finally studied using the exact solution of Lieb and Wu and the mean-field Hartree–Fock approximation. It is shown that the exact solution of the effective Hubbard model predicts a smooth crossover from the large polaron to a small polaron state while the mean field treatment leads to a discontinuous transition in the adiabatic regime.

Nature of the self-trapping transition in a one-dimensional Holstein?Hubbard model

The nature of the self-trapping transition in a many-polaron system is investigated within the framework of an extended Holstein–Hubbard model in one dimension using a variational method. The phonon degrees of freedom are first eliminated employing a modified Lang–Firsov transformation and an onsite squeezing transformation and then using a correlated squeezed phonon state to yield an effective Hubbard model which is finally studied using the exact solution of Lieb and Wu and the mean-field Hartree–Fock approximation. It is shown that the exact solution of the effective Hubbard model predicts a smooth crossover from the large polaron to a small polaron state while the mean field treatment leads to a discontinuous transition in the adiabatic regime.

Role of the superexchange interaction in magnetic transition and polaron crossover

The Hubbard–Holstein model is studied including double-exchange interaction and superexchange interaction using a variational phonon basis obtained through the modified Lang–Firsov transformation followed by the squeezing transformation. The kinetic energy, polaron crossover and magnetic transition are investigated as a function of electron–phonon (e–ph) coupling and electron concentration for different values of antiferromagnetic superexchange interaction ( J) between the core spins. The polaron crossover, magnetic transition and the suppression of ferromagnetic transition with J are discussed for the model.

Two-site two-electron generalized Hubbard–Holstein model: a perturbation study

The two-site two-electron generalized Hubbard–Holstein model is studied within a perturbation method based on a variational phonon basis obtained through the modified Lang–Firsov (MLF) transformation. The ground-state wave function and the energy are found including up to the seventh and eighth order of perturbation, respectively. The convergence of the perturbation corrections to the ground state energy, as well as to the correlation functions, are investigated. The kinetic energy and the correlation functions involving charge and lattice deformations are studied as a function of electron–phonon(e–ph) coupling and electron–electron interactions for different values of the adiabaticity parameter. The simultaneous effect of the e–ph coupling and Coulomb repulsion on the kinetic energy shows interesting features.

Perturbative expansion using a variational phonon basis for the Holstein model

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. A perturbation method based on this MLF basis is employed to calculate the single-electron ground-state energy and the dispersion of the polaronic band. The ground-state ( k=0) 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.

A CONVERGENT PERTURBATION METHOD FOR THE HOLSTEIN MODEL AND ITS APPLICATION TO THE DOUBLE EXCHANGE SYSTEM

We look for an analytical method which can describe the Holstein model over the entire region of electron–phonon (e–ph) coupling strength. Considering a toy model of two sites the perturbation expansions are developed for the Holstein Hamiltonian using different standard phonon bases. It is found that the modified Lang Firsov phonon basis shows satisfactory convergence for almost the entire range of e–ph coupling from the antiadiabatic limit to the intermediate range of hopping. This method is used to study the magnetic transition and the small to large polaron crossover in a double exchange model with e–ph interaction. Implications of the results in the context of manganites are discussed.

SMALL-TO-LARGE POLARON CROSSOVER IN ONE DIMENSION USING VARIATIONAL PHONON-AVERAGING TECHNIQUES

The ground-state properties of polarons in a one-dimensional chain is studied analytically within the modified Lang–Firsov (MLF) transformation using various phonon-averaging techniques. The object of the work is to examine how the analytical approaches may be improved to give rise to the real picture of polaronic properties as predicted by extensive numerical studies. The results are compared with those obtained from numerical analyses using the density matrix renormalization group (DMRG) and other variational techniques. It is observed that our results agree well with the numerical results particularly in the low and intermediate range of phonon coupling.

TWO-SITE POLARON PROBLEM: A PERTURBATION APPROACH WITH VARIATIONAL BASIS STATES

A convergent perturbation method using modified Lang Firsov transformation is developed for a two-site single-polaron system. The method is applicable for the entire range of the electron–phonon coupling strength from the antiadiabatic limit to the intermediate region of hopping. The single-electron energies, oscillator wavefunctions and correlation functions, calculated using this method, are in good agreement with the exact results.