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physica status solidi (b)

Self‐Consistent Normal Mode Analysis of the One‐Mode Polaron

S. L. Trubatch

Published Jan 1, 1973• 0 citations
Publisher

Abstract

Abstract The self‐energy of the one‐mode polaron model is calculated in both the weak and strong coupling regimes by using a variational principle which removes all linear Bose coupling terms and diagonalizes all quadratic Bose terms in the Hamiltonian. It is clearly demonstrated that this approximation scheme is not powerful enough to yield the correct strong coupling behavior.

DOI
10.1002/pssb.2220550136

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arXiv:cond-mat/0101162•Physical review. B, Condensed matter•Mar 12, 2002•40 citations
Abstract The self‐energy of the one‐mode polaron model is calculated in both the weak and strong coupling regimes by using a variational principle which removes all linear Bose coupling terms and diagonalizes all quadratic Bose terms in the Hamiltonian. It is clearly demonstrated that this approximation scheme is not powerful enough to yield the correct strong coupling behavior.
Abstract A general formalism for diagonalizing quadratic forms of boson operators [7], is used to obtain a translationally invariant variational ansatz which is suitable for calculating the polaron self‐energy, effective mass, and energy momentum relation in both coupling limits. The self‐energy and effective mass are calculated in the weak‐coupling limit. The strong coupling limit shows the wellknown α2‐dependence of the self‐energy. Diagonalizing the quadratic part of the recoil term gives rise to an additional self‐energy contribution to the cutoff.
Abstract A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.
Abstract A new translational invariant variational statement is given, which is exact to the order α2 for weak coupling and which is in the limiting case of strong coupling regarding the polaron self‐energy not poorer as‐0.105 α2 h ω. In the limiting case of weak coupling the polaron mass, the contributions to the self‐energy due to the individual components of Fröhlich's Hamiltonian, and up to the order α2 the mean value of virtual phonons in the ground state are exactly calculated.
A novel variational approach is presented for the calculation of the ground-state energy of the polaron in arbitrary N dimensions in the strong-coupling limit. By using the phonon coherent state to represent the wavefunction of phonons, a self-consistent integro-differential equation for the electron wavefunction is derived. The calculated results of the ground-state energy for N = 1, 2 and 3 agree well with the best results in the literature. It is also found that, for arbitrary N, the present results are less than the Feynman path integral ones by small percentages. It is proposed that this approach should be universal for systems involving polarons in the strong-coupling regime.
The main quasi-particle characteristics of the one-dimensional polaron are estimated within and beyond the most general Gaussian approximation at arbitrary electron-phonon coupling. We have derived explicitly the ground-state energy and the effective mass in the weak- and strong-coupling regimes. For arbitrary coupling, the Gaussian leading-order term of the polaron self energy improves the corresponding Feynman estimate and represents the lowest upper bound available. We have calculated the next non-Gaussian corrections. Taking into account systematically higher-order corrections does not perturb considerably the obtained results.
The paper deals with the ground and the first excited state of the polaron in the one-dimensional Holstein model. Various variational methods are used to investigate both the weak-coupling and strong-coupling case, as well as the crossover regime between them. Two of the methods, presented here, introduce interesting elements to the understanding of the nature of the polaron. Reliable numerical evidence is found that, in the strong-coupling regime, the ground and the first excited state of the self-trapped polaron are well described within the adiabatic limit. The lattice vibration modes associated with the self-trapped polarons are analyzed in detail, and the frequency softening of the vibration mode at the central site of the small polaron is estimated. It is shown that the first excited state of the system in the strong-coupling regime corresponds to the excitation of the soft phonon mode within the polaron. In the crossover regime, the ground and the first excited state of the system can be approximated by the anticrossing of the self-trapped and the delocalized polaron state. In this way, the connection between the behavior of the ground and the first excited state is qualitatively explained.