We study the Hubbard-Holstein model using slave-boson mean-field and a variational Lang-Firsov transformation. We identify weak and strong
e-ph coupling solutions, whose stability depends both on the bare
e-ph coupling and on the correlation strength. At mean field level the evolution from weak to strong electron-phonon coupling occurs via a first-order polaronic transition if the adiabatic parameter is below a critical value. In the strongly correlated regime and in the adiabatic limit, the region in which the weak-coupling solution is stable is sizeably enlarged with respect to the weakly correlated system and the Mott metal-insulator transition is found to be robust with respect to
e-ph interaction.
Polaronic features of dopant-induced charge carriers are manifest in recent experiments on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$
SrxCuO4+y and ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$
NixCuO4+y. To study the formation of hole (bi)polarons in such systems exhibiting strong Coulomb interactions, the Holstein t-J model is examined by means of a variational diagonalization technqiue for a wide range of phonon frequencies and electron-phonon (EP) couplings on finite square lattices, up to 18 effective sites in size. Including static displacement field, polaron, and squeezing effects, our approach allows for the description of nearly free polarons in the weak-coupling limit as well as for adiabatic Holstein polarons and nonadiabatic Lang-Firsov polarons in the case of strong EP interactions. At low doping levels and low phonon frequencies we demonstrate that antiferromagnetic spin correlations and EP interactions reinforce each other to the effect of lowering the threshold for polaronic self-localization in a strongly distorted lattice. This is contrasted with the nonadiabatic regime, where we observe the formation of Lang-Firsov polarons with moderate polaronic mass enhancement in a nearly undistorted lattice. In the case of two doped holes, the hole binding energy is analyzed in detail and we find that hole binding is enhanced as a dynamical (static) effect of a rather weak (extremely strong) EP interaction for delocalized (self-trapped) polarons. At quarter-filling we notice, in the adiabatic regime, a sequence of transitions as the EP coupling increases from nearly free mobile polarons to a polaronic superlattice and finally to a charge-separated state. The ordering of polarons disappears at higher phonon frequencies.
We solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.