For an electron interacting with harmonic dispersive phonons in a periodic lattice, the possibility of wave (soliton)-like solutions of the generalised master equations (GME) for the electron site occupation probabilities is investigated. For the current form of the small-polaron Hamiltonian, the long-time asymptotics of the memory functions (kernels of the GME) in the small-polaron basis are discussed to all orders in the electron hopping integrals.Using this result, one obtains (for the initially localised electron) the t 2 -behaviour of the mean-square electron displacement at zero temperature. This corresponds to existing soliton solutions of the Schrodinger equation. At non-zero temperatures, the wave solution is damped.
For an electron interacting with harmonic dispersive phonons in a periodic lattice, the possibility of wave (soliton)-like solutions of the generalised master equations (GME) for the electron site occupation probabilities is investigated. For the current form of the small-polaron Hamiltonian, the long-time asymptotics of the memory functions (kernels of the GME) in the small-polaron basis are discussed to all orders in the electron hopping integrals.Using this result, one obtains (for the initially localised electron) the t 2 -behaviour of the mean-square electron displacement at zero temperature. This corresponds to existing soliton solutions of the Schrodinger equation. At non-zero temperatures, the wave solution is damped.
The theory of Wilson's solitary-wave acoustic polaron motion is generalized to include the effects of quantum fluctuations and the non-adiabatic behaviour of the electron-phonon coupling. The polaron dynamics are described quantum mechanically in terms of master equations for the electronic variables. As a final result, it emerges that due to the considered quantum effects for v<<c (v=polaron velocity, c=sound velocity) the polaron behaviour is determined by the well-known semiclassical non-linear Schrodinger equation, whereas for v<c the polaron motion is overdamped but shows by no means a divergent behaviour as obtained in semiclassical theories. In addition, for v=c the phonon cloud around the electron disappears and the electron motion becomes diffusive.
We introduce a variational wave function to study the polaron formation when the electronic transfer integral depends on the relative displacement between nearest-neighbor sites giving rise to a nonlocal electron-phonon coupling with optical phonon modes. We characterize the polaron crossover by analyzing ground-state properties such as the energy, the electron-lattice correlation function, the average phonon occupation, and the quasiparticle spectral weight. Variational results are found in good agreement with numerical exact diagonalization of small clusters,and follow the correct perturbative result at weak coupling. We determine the polaronic phase diagram and we find that the tendency towards strong localization is hindered from the pathological sign change of the effective next-nearest-neighbor hopping.
The structure of the translationally-invariant diagrammatic perturbation theory for one polaron is examined on the 1D discrete lattice described by the Holstein Hamiltonian. The latter is characterized by the electron hopping t, the phonon frequency ω_0 and the electron-phonon coupling g. It is shown that the polaron localization (and translation) properties are contained in the electron propagator of one electron, intermittently added to the lattice, and/or in the phonon correlation function with one electron permanently present in the lattice. The order by order analysis in g/ω_0 shows that the expansion of the irreducible electron self-energy corresponds to the expansion of the phonon correlation function, rather than of the irreducible phonon self-energy. The range of polaronic correlations is determined in this way. For small t/ω_0 and already to the second order g/ω_0 small, the electron-lattice correlation becomes very short ranged, i.e. the polaron is already localized to one site, although the overall translational symmetry remains unbroken. For large t/ω0, the second order result is meaningful up to large g/ω_0 ≈ (t/ω_0)^\frac14, where it becomes degenerate with the results for the large adiabatic Holstein polaron. This suggests that the translationally invariant perturbation theory crosses then over smoothly, without symmetry breaking, into the adiabatic, continuous quantum limit, as rigorously demonstrated in the companion paper. Thus the quantum theory of the large adiabatic Holstein polaron provides a simple, instructive example of the quantum crossover which replaces the behavior in the quantum critical point.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
We present results for the time evolution of a one-dimensional system consisting of an electron, described by a tight-binding Hamiltonian and a harmonic latttice, coupled by a deformation-type potential. We solve numerically the nonlinear system of equations of motion for this model in order to study the effects of varying the electronic effective mass for several initial conditions and coupling strengths. Different types of localized and extended states are formed with features that are absent from the traditional polaronic states and depend very strongly on the initial electronic configuration and effective mass in a very often unexpected manner. We find that, in general, an increase of the initial electronic energy decreases the ability of the system to form localized states. However, a large effective mass favors localized polaron formation for initially localized electrons, but this is not always the case for initially extended electronic states. In the latter case, increasing the effective mass of an electron initially close to the bottom of the band makes localization more difficult, while for an initially highly excited electron, localized polaron formation is possible only when the electronic effective mass and the atomic masses of the lattice become of the same order. Finally, for a small parameter range, we find an impressive recurrence, a periodic and a complete exchange between the electronic and vibrational degrees of freedom of a small part of the initial electronic energy.
A density-matrix treatment of small-polaron motion is presented for the case in which the electronic overlap term of the total Hamiltonian is a small perturbation. The principal result of the density matrix formalism is that total small-polaron mobility can be expressed as the sum of a band part $〈{{v}_{o}}^{2}\ensuremath{\tau}〉$, characteristic of the low-temperature regime (T<Tt), plus a part (WTa2) describing the hopping motion dominant at high temperatures (T>Tt). This verifies the separation of the above two types of motion made on the basis of physical arguments. In addition, the present treatment avoids certain formal divergences in the integrals for the jump probabilities. Furthermore, the hopping contribution does not require localization of the polaron at a particular site, but follows from a translationally invariant formulation. These results are obtained, in part, from lowest order Boltzmann equations which are derived both in the local-site and polaron-band representations. The principal contributions to the scattering terms of the Boltzmann equations are determined by interference effects between the matrix elements, which are examined in some detail.