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physica status solidi (b)

Nonadiabatic Electron–Phonon Interaction. Matrix Continued Fraction Treatments and Reduction of High‐Dimensional Problems

A. Scheuing, P. Reineker, C. Durst, E. Sigmund

Published Jul 1, 1987• 5 citations
Publisher

Abstract

Abstract An electronic two‐level system interacting with two different vibrational modes via displacive and transitive couplings is investigated. The eigensolutions are determined using a matrix continued fraction method including infinite matrices. The approximations used in the numerical evaluation are discussed. As another example for the applicability of the continued fraction method it is shown that the E–e‐Jahn‐Teller Hamiltonian proves to be a specialized case, whose eigenvalue problem may be solved using ordinary continued fractions. Finally, the eigenvalue problem of the extended Hubbard Hamiltonian of a dimer with linear local electron–phonon coupling is reduced to the evaluation of a matrix continued fraction with three‐dimensional matrices.

DOI
10.1002/pssb.2221420118

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•Aug 1, 1988•0 citations

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D. Schneider, K.‐H. Höck, K. Ziegler

The properties of a dilute electron gas, coupled to the lattice degrees of freedom, are studied and compared with the properties of an electron gas at half-filling, where spinless fermions with two orbitals per lattice site are considered. The simplest model which includes both the local electron-lattice interaction of the Jahn-Teller type and the electronic correlations is the E ⊗ β-Jahn-Teller-Hubbard model. We analyze the formation and stability of Jahn-Teller polarons and bipolarons, respectively. Our approach is based on a hopping expansion in the strong-coupling regime. The results are compared with recently published findings for the Hubbard-Holstein model [1, 2]. The special case of the Jahn-Teller-Hubbard model at half-filling is mapped on a spin-1/2 Heisenberg model with phonon-dependent coupling constants. This has been derived within a projection formalism that provides a continued-fraction representation of the Green's function. We study the exact solution for two and three particles and compare it with the effective theory on the infinite lattice with one particle per site. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

arXiv:cond-mat/0408180•Physica status solidi. C, Conferences and critical reviews/Physica status solidi. C, Current topics in solid state physics•Nov 1, 2004•1 citation
Abstract An electronic two‐level system interacting with two different vibrational modes via displacive and transitive couplings is investigated. The eigensolutions are determined using a matrix continued fraction method including infinite matrices. The approximations used in the numerical evaluation are discussed. As another example for the applicability of the continued fraction method it is shown that the E–e‐Jahn‐Teller Hamiltonian proves to be a specialized case, whose eigenvalue problem may be solved using ordinary continued fractions. Finally, the eigenvalue problem of the extended Hubbard Hamiltonian of a dimer with linear local electron–phonon coupling is reduced to the evaluation of a matrix continued fraction with three‐dimensional matrices.
A recursive method is developed to treat electrons coupled to phonons. It is applied to small systems with $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller coupling. Two cases are considered, a model with one electron and two orbitals on a single site (related to the Rabi Hamiltonian) and a model with two electrons on two sites. The corresponding Green's functions are represented by rational functions. It is found that the spectra change substantially when one phonon couples to the electron but are relatively robust under an increasing number of phonons.
For both identical and distinguishable particles and for an arbitrary polynomial approximation to the realistic two-body forces, the A-body Hamiltonian may be converted into an infinite block-three-diagonal matrix in the properly arranged, translationally invariant, oscillator basis. As a consequence, the exact Green function and all the projections of eigenstates of the microscopic Schrodinger equation become expressible in terms of the matrix continued fractions. This generalises the recent reformulation of the one-dimensional A=1 anharmonic problem by Graffi and Greechi (1975). The quick convergence of this non-perturbative method of solving the many-body bound-state problem is demonstrated for the simplest three- and four-body examples. With the core-possessing type V(r)=-r 2 +r 4 of the spin- and isospin-independent two-body interaction, only the 10 -3 error in the three-bosonic ground-state energy arises from restricting the formulae to 5*5 dimensional matrices.
The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrons and periodic boundary conditions. A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal-acoustic phonons through a bilinear interaction. The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes. In the adiabatic (large-ionic-mass, small-vibrational-frequency) limit there is an analytical solution which indicates that the transition between the nondistorted (weak-coupling) and the distorted (strong-coupling) phases can be either continuous or discontinuous, depending on whether the value of electron-electron interaction is smaller or larger than a critical value. In the extremely-high-frequency limit (ionic mass M\ensuremath{\rightarrow}0) it is also possible to find an analytical solution (for M=0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction. In the intermediate case (finite, nonvanishing M), numerical solutions exhibit a continuous transition from the nondistorted to the distorted phase.
The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrans and periodic boundary conditions.A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal acoustic phonons through a bilinear interaction.The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes.In the adiabatic (large ionic mass, small vibrational frequency) limit there is an analytical solution which indicates that the transition between the non-distorted (weak coupling) and the distorted (strong coupling) phases can be either continuous or discontinuous, depending on whether the value of the electron-electron interaction is larger or smaller than a critical value.In the extremely high-frequency limit (ionic mass M ~ 0 ) it is also possible to find an analytical solution (for M = 0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction.In the intermediate case (finite, nonvanishing M) numerical solutions exhibit a continuous transition from the non-distorted to the distorted phase.
The properties of a dilute electron gas, coupled to the lattice degrees of freedom, are studied and compared with the properties of an electron gas at half-filling, where spinless fermions with two orbitals per lattice site are considered. The simplest model which includes both the local electron-lattice interaction of the Jahn-Teller type and the electronic correlations is the E ⊗ β-Jahn-Teller-Hubbard model. We analyze the formation and stability of Jahn-Teller polarons and bipolarons, respectively. Our approach is based on a hopping expansion in the strong-coupling regime. The results are compared with recently published findings for the Hubbard-Holstein model [1, 2]. The special case of the Jahn-Teller-Hubbard model at half-filling is mapped on a spin-1/2 Heisenberg model with phonon-dependent coupling constants. This has been derived within a projection formalism that provides a continued-fraction representation of the Green's function. We study the exact solution for two and three particles and compare it with the effective theory on the infinite lattice with one particle per site. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)