Based on the exact cluster diagonalization and recent Quantum Monte Carlo simulations we analyze dynamic properties of small polarons and bipolarons formed by short-range (Holstein) and long-range (Fröhlich) electron-phonon interactions. We show that the exact results agree well with canonical Holstein theory for a cluster and with Lang-Firsov theory for a lattice. Lang-Firsov theory of a single polaron and our $1/\lambda$ perturbation expansion for a multipolaron system are practically exact in a wide range of the adiabatic parameter $\omega /t$ and the electron-phonon coupling λ for a long-range interaction. (Bi)polarons exist in the itinerant Bloch states at temperatures below the characteristic phonon frequency no matter which values the parameters of the system take. We show that recent claims by several authors with regards to a breakdown of Holstein-Lang-Firsov theory of a small polaron and the “impossibility” of bipolaronic superconductivity are the result of an erroneous interpretation of the electronic energy levels of the two-site Holstein model and a misunderstanding of the electron-phonon interaction in ionic solids with polaronic carriers. A “phase” diagram in $t/\omega -\lambda$ space is proposed to elucidate the BCS and (bi)polaronic domains. Bipolaron theory provides a parameter-free expression for the superconducting critical temperature of layered cuprates. Crystallization of the (bi)polaronic liquid is shown to be impossible in the range of the parameters typical for cuprates. The small Fröhlich polaron has spectral features compatible with single-particle tunneling and photoemission in cuprates.

• Received 13 January 1999

DOI:https://doi.org/10.1103/PhysRevB.61.12315