 
FIZIKA A 14 (2005) 2, 153  166
POLARONS BY TRANSLATIONALLY INVARIANT DIAGRAMMATIC PERTURBATION THEORY
OSOR S. BARIŠIĆ^{a} and SLAVEN BARIŠIĆ^{b}
^{a}Institute of Physics, Bijenička c. 46, HR10 000
Zagreb, Croatia
^{b}Department of Physics, Faculty of Science, University of Zagreb, Bijenička c.
32
HR10 000 Zagreb, Croatia
Received 15 March 2005; Accepted 26 September 2005
Online 7 November 2005
The structure of the translationallyinvariant 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 w_{0} and the electronphonon 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/w_{0} shows that the expansion of
the irreducible electron selfenergy corresponds to the expansion of the phonon
correlation function, rather than of the irreducible phonon selfenergy. The range of
polaronic correlations is determined in this way. For small t/w_{0}
and already to the second order g/w_{0} small, the
electronlattice 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/w_{0}, the second order result is meaningful up
to large g/w_{0} » (t/w_{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.
PACS numbers: 71.38.k, 63.20.Kr
UDC 534.16
Keywords: polaron, electronphonon coupling, Holstein model, symmetry breaking
