FIZIKA A 14 (2005)  2, 153 - 166

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aInstitute of Physics, Bijenička c. 46, HR-10 000 Zagreb, Croatia
bDepartment of Physics, Faculty of Science, University of Zagreb, Bijenička c. 32
HR-10 000 Zagreb, Croatia

Dedicated to the memory of Professor Vladimir Šips

Received 15 March 2005; Accepted 26 September 2005
Online 7 November 2005

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 w0 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/w0 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/w0 and already to the second order g/w0 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/w0, the second order result is meaningful up to large g/w0 » (t/w0)\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.

PACS numbers: 71.38.-k, 63.20.Kr
UDC 534.16

Keywords: polaron, electron-phonon coupling, Holstein model, symmetry breaking

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