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Physical review. B./Physical review. B

Exact solution for the ground-state energy band of an extended one-dimensional Holstein model

Feng Pan, L. R. Dai, Jerry Paul Draayer

Published Nov 13, 2024• 2 citations

Abstract

We demonstrate that the ground-state energy band of an extended one-dimensional Holstein model, featuring a specific type of nonlinear electron-phonon coupling for a single electron in the band, is exactly solvable. Numerical results indicate that, despite the significant differences in electron-phonon coupling between the extended Holstein model and the original Holstein model, the ground-state energy of the Holstein model can be accurately reproduced by the extended model through appropriate rescaling of the coupling parameter. Additionally, for an $N$-site lattice, there are ${2}^{N\ensuremath{-}1}$ distinct choices for the set of parameters in the nonlinear electron-phonon coupling, all yielding the same ground-state energy but with varying ground-state properties. This flexibility allows for the selection of the optimal parameter set in the extended model to best simulate the ground-state properties of the system.

DOI: 10.1103/physrevb.110.205128

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We demonstrate that the ground-state energy band of an extended one-dimensional Holstein model, featuring a specific type of nonlinear electron-phonon coupling for a single electron in the band, is exactly solvable. Numerical results indicate that, despite the significant differences in electron-phonon coupling between the extended Holstein model and the original Holstein model, the ground-state energy of the Holstein model can be accurately reproduced by the extended model through appropriate rescaling of the coupling parameter. Additionally, for an

N

-site lattice, there are

{2}^{N\ensuremath{-}1}

distinct choices for the set of parameters in the nonlinear electron-phonon coupling, all yielding the same ground-state energy but with varying ground-state properties. This flexibility allows for the selection of the optimal parameter set in the extended model to best simulate the ground-state properties of the system.

We study the nonlinear lattice effect on the ground state in a one-dimensional spinless Holstein model with nonadiabitical coupling and squeezing by means of the parameter variational approach. Our results show that the introduction of a hard quartic term in the lattice potential increases the ground state energy of the system when electron-phonon coupling is strong, and the increment is sensitive to the magnitude of the lattice quartic force constant. In this case the nonlinear lattice effects should be taken into account to describe satisfactorily some physical properties of the coupling system.

We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.

The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is solved by using a Bethe ansatz in analogue to that for the one-dimensional spinless Fermi–Hubbard model. Excitation energies and the corresponding wavefunctions of the model are determined by a set of partial differential equations. It is shown that the model is, at least, quasi-exactly solvable for the two-site case, when the phonon frequency, the electron–phonon coupling strength and the hopping integral satisfy certain relations. As examples, some quasi-exact solutions of the model for the two-site case are derived.

A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5,1.1, and 2.1 (in units of phonon frequency ω 0 ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0,2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.

A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5, 1.1, and 2.1 (in units of phonon frequency ωo ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0, 2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.