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Physica B+C

Macroscopic phonon wave functions and the Fröhlich one-dimensional superconductor

J.W. Evans

Published Jul 1, 1976• 0 citations
Publisher
DOI
10.1016/0378-4363(76)90125-x

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A refined approach where the wave function is gradually improved is developed in the one-dimensional (1D) large-Frohlich-polaron problem. The results for the ground-state energy, the effective mass, and the average number of virtual phonons are obtained for a wide range of the coupling constant It, and are in agreement with those of the Feynman path-integral formalism. In addition, in the weak-coupling limit, the expansions of these observables in powers of the coupling constant are exactly calculated up to the alpha 3 -term. They are E=- alpha -0.06066 alpha 2 -0.00844 alpha 3 , m*=1+0.5 alpha +0.19194 alpha 2 +0.06912 alpha 3 , and N=0.5 alpha +0.12132 alpha 2 +0.029540 alpha 3 , which agree with the known results of the fourth-order perturbation theory.
Energy levels of two-dimensional Fr\"ohlich polarons in a magnetic field are calculated variationally by use of a generalization of the Lee-Low-Pines ansatz for the variational wave function allowing the phonon distribution function to depend on the electronic coordinates. It is shown that this choice leads to the correct interpolation between the weak- and strong-field limits. Comparing our ground-state energies with those obtained within second-order perturbation theory and the path-integral method, we argue that neither of these methods gives a satisfactory description of the dynamics of the electron-phonon interaction under the influence of an external field.
According to recent quantum Monte Carlo simulations, the small polaron theory is practically exact in a wide range of the long-range (Fröhlich) electron-phonon coupling and adiabatic ratio. Applying Lang-Firsov transformation, we derive a single-particle Green's function describing the propagation of small Fröhlich polaron. One- and two-dimensional spectral functions are studied by expanding Green's function perturbatively. Remarkably, the coherent spectral weight ( Z ) and effective mass ( Z ′) renormalisation exponents are found to be different with Z ′ ≫ Z , which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.