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

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.