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Russian Journal of Physical Chemistry B

Polaron on a Lattice with a Trap in the Su–Schrieffer–Heeger Approximation

Vladimir Nikolaevich Likhachev, George A Vinogradov

Published Mar 1, 2020• 1 citation
Publisher

Abstract

Abstract The wave function and energy of the ground state of a lattice described by the Su–Schrieffer–Heeger Hamiltonian with a trap were found using three different methods. In the first of these methods, the energy of the ground state was calculated by numerical minimization of the energy functional. In the second, an auxiliary Hamiltonian was constructed, the diagonalization of which determined the energy of the ground state and the wave functions. And in the third, the wave function is completely restored analytically from the asymptotics of the wave function. All the three approaches produced results that agreed with an accuracy of 10–15.

DOI
10.1134/s1990793120020098

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Abstract The wave function and energy of the ground state of a lattice described by the Su–Schrieffer–Heeger Hamiltonian with a trap were found using three different methods. In the first of these methods, the energy of the ground state was calculated by numerical minimization of the energy functional. In the second, an auxiliary Hamiltonian was constructed, the diagonalization of which determined the energy of the ground state and the wave functions. And in the third, the wave function is completely restored analytically from the asymptotics of the wave function. All the three approaches produced results that agreed with an accuracy of 10–15.
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 obtain analytical expressions for the large- and small-radius polarons on the one-dimensional lattice in the TBA approximation. The equations of motion for this model are treated classically for the oscillator subsystem, while a quantum description is used for the electron. The electron-phonon interaction is considered in the linear Su--Schrieffer--Heeger approximation. Good agreement between analytical formulae and accurate numerical simulation is obtained. The dynamics of polaron formation from different initial conditions is considered. Some features of the wave function evolution, governed by the finite lattice length, are elucidated.
The method of moments is employed to determine the ground state of the phonon-polaron system. The trial function employed in this calculation is a bare-electron wave function. It appears that the method is convergent for this problem. Although the energy is in general a complicated function of the parameter and must be determined numerically, it is possible to formally construct a power series expansion for the energy about the unperturbed energy in powers of the coupling constant. The first term in this series has been determined and agrees with the result of conventional perturbation theory. The second term has not yet been evaluated to all orders but it appears that it will have the opposite sign from the corresponding term in perturbation theory. The second and higher terms in some orders of approximation are singular for small values of the cutoff.
The many-body ground state of a very general class of electron-phonon Hamiltonians is proven to contain a spin singlet (for an even number of electrons on a finite lattice). The phonons interact with the electronic system in two different ways; there is an interaction with the local electronic charge and there is a functional dependence of the electronic hopping Hamiltonian on the phonon coordinates. The phonon potential energy may include anharmonic terms, and the electron-phonon couplings and the hopping matrix elements may be nonlinear functions of the phonon coordinates. An attractive Hubbard-type on-site interaction may also be added. If the hopping Hamiltonian is assumed to have no phonon-coordinate dependence, then the ground state of a finite system is also shown to be unique, implying that there are no ground-state level crossings, and that the ground-state energy is an analytic function of the parameters in the Hamiltonian. In particular, in a finite system any self-trapping transition is a smooth crossover not accompanied by a nonanalytical change in the ground state. The spin-singlet theorem applies to the Su-Schrieffer-Heeger model and both the spin-singlet and uniqueness theorems apply to the Holstein and attractive Hubbard models as special cases. These results hold in all dimensions---even on a general graph without periodic lattice structure.
The ground-state energy of the optical polaron is calculated by the generalized path-integral formulation. The ground-state approximation is made to simplify the complicated expression, and the Ritz variational and direct integration methods are used to obtain the ground-state energy in the whole range of coupling strength. The results agree with previous work, and it is found that there is a transition for coupling constant ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}9.2$. The ground-state energy obtained by harmonic approximation, which is equivalert to taking Gaussian-like trial wave functions, is compared with those obtained by Pekar's trial wave function.