We analyse the validity ranges of the adiabatic and anti-adiabatic approximations for a model with electron - electron and electron - phonon interactions for a three-site cluster. This model describes the motion of apical oxygen atoms in . We show that the adiabatic approximation still applies for critical infrared coupling where the gap between the ground state and first excited electronic state becomes nearly equal to the double-well potential barrier. For this high coupling value, the motion of holes and ions is already highly correlated. We arrive at the non trivial conclusion that in this model the adiabatic approximation is valid even in a parameter region where the electronic energy scale is not much larger than the phononic energy scale. This approximation is able to describe polaronic effects which have been shown to appear in the infrared absorption spectrum in previous exact numerical calculations. We also investigate the range of validity of the anti-adiabatic approximation. We find that the window where none of the above-mentioned approximations is valid shrinks with decreasing on-site repulsion U until it closes at U = 0 in agreement with other work in the literature.
We analyse the validity ranges of the adiabatic and anti-adiabatic approximations for a model with electron - electron and electron - phonon interactions for a three-site cluster. This model describes the motion of apical oxygen atoms in . We show that the adiabatic approximation still applies for critical infrared coupling where the gap between the ground state and first excited electronic state becomes nearly equal to the double-well potential barrier. For this high coupling value, the motion of holes and ions is already highly correlated. We arrive at the non trivial conclusion that in this model the adiabatic approximation is valid even in a parameter region where the electronic energy scale is not much larger than the phononic energy scale. This approximation is able to describe polaronic effects which have been shown to appear in the infrared absorption spectrum in previous exact numerical calculations. We also investigate the range of validity of the anti-adiabatic approximation. We find that the window where none of the above-mentioned approximations is valid shrinks with decreasing on-site repulsion U until it closes at U = 0 in agreement with other work in the literature.
The phase diagram of a two-dimensional N-site N-electron system (N\ensuremath{\gg}1) with site-diagonal electron-phonon (e-ph) coupling is studied in the context of polaron theory, so as to clarify the competition between the superconducting (SC) state and the charge-density wave (CDW) state. The Fermi surface of noninteracting electrons is assumed to be a complete circle with no nesting-type instability in the case of weak e-ph coupling, so as to focus on such a strong coupling that even the standard ``strong-coupling theory'' for superconductivity breaks down. Phonon clouds moving with electrons as well as a frozen phonon are taken into account by a variational method, combined with a mean-field theory. It covers the whole region of three basic parameters characterizing the system: the intersite transfer energy of electron T, the e-ph coupling energy S, and the phonon energy \ensuremath{\omega}. The resultant phase diagram is given in a triangular coordinate space spanned by T, S, and \ensuremath{\omega}. In the adiabatic region \ensuremath{\omega}\ensuremath{\ll}(T,S) near the T-S line of the triangle, each electron becomes a large polaron with a thin phonon cloud, and the system changes discontinuously from the SC state to the CDW state with a frozen phonon as S/T increases. In the inverse-adiabatic limit \ensuremath{\omega}\ensuremath{\gg}(T,S) near the \ensuremath{\omega} vertex of the triangle, on the other hand, each electron becomes a small polaron, and the SC state is always more stable than the CDW state, because the retardation effect is absent. Thus, the polaron radius decreases and the SC region expands in the triangle as T/\ensuremath{\omega} decreases. It is found, for the first time, that the energy gap of the SC state for a given T and S becomes maximum at the intermediate region \ensuremath{\omega}\ensuremath{\sim}T, indicating the importance of the polaron effect. The collective excitation within the gap of the SC state is also studied by the random-phase approximation, and is found to change its nature continuously from the pair-breaking type to the superfluid type as S/T increases.
We give an analytical formula in term of continued fraction expansions for the spectral function of a tunnelling electron, coupled to a local lattice oscillation, in a two-site cluster at non-zero temperature. We also study the spectral function of the polaron, a better defined quasi-particle in the anti-adiabatic regime and at sufficiently low temperature. The exact results obtained allow us to look into a wide range of temperature and coupling. Asymptotic results can be obtained directly from the continued fraction expansions in both adiabatic and anti-adiabatic regimes. In the intermediate/strong anti-adiabatic case, in contrast to the usual Lang-Firsov approximation scheme, we found that there is no shrinking of the polaron band as temperature increases. Polaron bandwidth gets broader by temperature effects.
We solve a model of interacting electrons coupled to longitudinal phonons using the density matrix renormalization group method. The model is parametrized for trans-polyacetylene. We calculate the ground state and first excited odd-parity singlet and triplet states. We investigate their energies and geometries for up to 102 sites. The transition energy and the soliton width of the triplet state show significant deviations from the adiabatic approximation for chain lengths larger than the classical soliton size. In contrast, the transition energy of the singlet is close to the adiabatic prediction.
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 phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.