We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov We discuss in particular the case for which the on-site Coulomb repulsion is exactly cancelled by the phonon-mediated attractive interaction and suggest that polarons exchanging phonon clouds can lead to polaron pairing and superconductivity. It is then the frequency of the collective mode of phonon clouds being larger than the bare frequency, which determines the superconducting transition temperature.The aim of the present paper is to gain further insight into the mutual influence of strong onsite Coulomb repulsion and strong electron-phonon interaction by using the single-band HubbardHolstein model and a recently developed diagrammatic approach 1-4. For simplicity we consider coupling to dispersionless phonons only, although this might not be the most interesting case with respect to superconductivity. However, previous investigations 5-7 have shown that the HubbardHolstein model 8-9 constitutes a formidable problem of its own. Other authors have also intensively studied this model Hamiltonian 10-13.Because the interactions between electrons and electrons and phonons are strong, we include the Coulomb repulsion in the zero-order Hamiltonian and apply the canonical transformation of Lang and Firsov 14 in order to eliminate the linear electron-phonon interaction. In the strong electronphonon coupling limit the resulting Hamiltonian of hopping polarons (i.e., hopping electrons surrounded by clouds of phonons) can lead to an attractive interaction among electrons being mediated by the phonons. In this limit the chemical potential, on-site Coulomb energy as well as the frequency of the collective mode of phonon clouds (which is much larger than the bare frequency of the Einstein oscillators) are strongly renormalized 7,15,16 affecting the dynamical properties of the polarons and the character of the superconducting transition. This will be discussed by assuming that renormalized on-site Coulomb repulsion and attractive electron-electron interaction completely cancel each other. We suggest that the resulting superconducting state with polaronic Cooper pairs is mediated by the exchange of phonon clouds during the hopping processes of the electrons.The initial Hamiltonian of correlated electrons coupled to optical phonons with bare frequency 0
The possibility is indicated of applying the theory of superconductors with overlapping energy bands to describe the thermodynamic and electromagnetic properties of the high-temperature compounds La 2−x (Ba, Sr) x CuO 4 and Y Ba 2 Ci 3 O 7−δ . The twoband model was used to obtain high values of T c , two energy gaps 2∆ 1 /T c > 3.5 and 2∆ 2 /T c < 3.5, large negative values of d ln T c /d ln V (V is the volume) in lanthanum ceramics, small values of the jump in the electron heat capacity at T = T c , negative curvature of the upper critical magnetic field H c2 near the transition temperature, etc. Such behavior of the above quantities is observed experimentally. A description is also obtained of the decrease in T c as the disordering of oxygen increases, and also as copper atoms are replaced by a nonmagnetic impurity (Al, Zn, etc.). The main mechanism responsible for this decrease is the interband scattering of electrons by impurities and by randomly distributed oxygen vacancies. A theory has been developed of multiband superconductors which takes into account the points of high symmetry in momentum space. On the basis of this theory one can explain the existence of a plateau in the dependence of T c on δ for Y Ba 2 Cu 3 O 7−δ , and also in the dependence of T c on x for La 2−x (Ba, Sr) x CuO 4 , that has been observed in a number of experiments. Moreover, this theory also explains the presence of two maxima in the dependence of T c on pressure for Bi 2 Sr 2 CaCu 2 O 8 .
We exhibit, for arbitrary time variations of the parameters of the generalised harmonic oscillator Hamiltonian, a canonical transformation which facilitates an exact analysis of quantal phase and classical angle. Formulae for Berry's phase and Hannay's angle are readily obtained by taking the adiabatic limit of the exact theory.
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