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Civil War Book Review

On Solutions of the One-dimensional Holstein Model

Feng Pan, Jerry Paul Draayer

Published Dec 1, 2006• 0 citations
Publisher

Abstract

The one-dimensional Holstein model of spinless or hard-core fermions interacting with dispersionless phonons is proved to be exactly solvable. Excitation energies and the corresponding wavefunctions of the model are obtained by using a simple extended Bethe ansatz.
DOI
10.1142/9789812772527_0034

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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.
The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various low-energy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the antiadiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
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 NNN-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.