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International Journal of Modern Physics B

Analytical Solutions of the Two-Site Holstein Model by the Coherent State Expansion

Zijing Lin, Rongsheng Han, Wang Kelin

Published Sep 30, 2003• 2 citations
Publisher

Abstract

The polaron problem presents a highly nonlinear dynamical system in which the charge and lattice deformations are intricately coupled together. The coherent state expansion method was applied to the two-site Holstein model. Analytical solutions are found for the first, the second and the third order approximations. Comparison is made among the exact numerical results, based upon solving a recursive relation for the expansion coefficients, and the analytical solutions up to the third order approximation for various e-ph coupling strengths and electron hopping parameters.

DOI
10.1142/s0217979203022271

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The polaron problem presents a highly nonlinear dynamical system in which the charge and lattice deformations are intricately coupled together. The coherent state expansion method was applied to the two-site Holstein model. Analytical solutions are found for the first, the second and the third order approximations. Comparison is made among the exact numerical results, based upon solving a recursive relation for the expansion coefficients, and the analytical solutions up to the third order approximation for various e-ph coupling strengths and electron hopping parameters.
The two-site Holstein model is studied by the analytical method of the coherent-state expansion. We obtain the recursive relationship among the expansion coefficients, which leads to the exact solutions of the eigenstate properties. The exact ground-state results in the whole coupling strength region and from antiadiabatic to adiabatic limits are in perfect agreement with that available by the exact numerical diagonalization technique. Comparison with the best perturbation results shows the success and shortcoming of the perturbative method. The coherent-state expansion method is a promising approach in studying electron-phonon $(e\ensuremath{-}\mathrm{ph})$ coupling systems.
We obtain the exact solution of energy spectrum and dynamics for the two-site Hubbard-Holstein model by the coherent states orthogonal expansion method. The influences of coupling strength g, the average number of phonons n and the initial electronic state on the evolution of system entanglement and von Neumann entropy are discussed. Numerical results are as follows. (a) Entanglement evolution with time shows a good periodicity. When the other parameters are fixed, the evolution period decreases as the coupling strength g goes up but it is independent of the average number of phonons n. (b) The von Neumann entropy of the system demonstrates strict synchronia with the electronic state occupancy probability. (c) Under the weak coupling strength and low average number of phonons, the initial electronic state c2+ c2+|Oe or c1+ c1+|Oe shows larger maximum von Neumann entropy during its evolution than that of c1+c2+c1+c2++ but they gradually approach to each other with the increase of coupling strength or average number of phonons.
Abstract The Holstein Hamiltonian was proposed half a century ago; since then, decades of research have come up empty handed in the pursuit of a closed-form solution. An exact solution to the two-site Holstein model is presented in this paper. The obtained results provide a clear image of the Hamiltonian structure and allow for the investigation of the symmetry, energy level crossings and polaronic characteristics of the system. The main mathematical tool is a three-term recurrence relation between the wave function amplitudes, which was obtained using the properties of a family of orthogonal functions, namely the Poisson–Charlier polynomials. It is shown that, with the appropriate choice of basis, the eigenfunctions of the problem naturally fall into two families (parities) associated with the discrete Z 2 symmetry of the Hamiltonian. The asymptotic solution to the recurrence relation is found by using the Birkhoff expansion. The asymptotic sets the truncation criterion for the wave function, which ensures the accurate calculation of the energy levels for any strength of electron–phonon interaction. The level crossing of states with different parities is discussed and the exact points of broken symmetry are found analytically. The results are used as the building blocks for studying a four-site system. The inherited symmetries lead to the formation of a sparse matrix that is convenient for numerical calculations.
A recently proposed variational approach based on an expansion in coherent states is used to investigate the single polaron features in the one-dimensional Holstein model. The method provides highly accurate results, both for static and dynamic properties, over a wide range of the polaron parameter space, from weak to strong coupling regimes, with a little numerical effort when compared with standard numerical techniques. The results for the spectral weight function and the real part of the optical conductivity are reported in the moderate adiabatic regime. We show that the spectral function exhibits a strong dependence on the polaron wave number kkk that is not present in the standard analytical approaches. The optical conductivity, in the strong coupling regime, displays a high-energy peak due to the photoionization of the small polaron, while, at lower energies, there is evidence of electronic transitions between localized states. These results show that the coherent-state basis is a very powerful basis to study interacting electron-phonon systems.
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.