This chapter describes some of the most important theories of the strong-coupling polaron. The nature of polarons in the strong-coupling limit is in some sense qualitatively different from the weak or intermediate-coupling polaron—the localization of a polaron being a case in point. Though a qualitative picture of a strong-coupling polaron is easy to visualize as opposed to the weak-coupling polaron, the mathematical formulation of a strongly coupled polaron is rather a difficult problem from the quantum mechanical point of view. In fact, the exact polaron energy in the weak-coupling limit can be obtained from the Frohlich Hamiltonian analytically, while the same is not possible in the strong-coupling limit. In the strong-coupling case, the electron velocity is much larger than the phonon frequency and therefore electron can follow the ion motion adiabatically. The adiabatic expansion of the polaron energy in the strong-coupling regime was reported by Bogoliubov and Tiablikov.
Using the Lanczos method in linear chains we study the double exchange model in the low concentration limit, including an antiferromagnetic super-exchange K. In the strong-coupling limit we find that the ground state contains ferromagnetic polarons whose length is very sensitive to the value of
K/t. We investigate the dispersion relation, the trapping by impurities, and the interaction between these polarons. As the overlap between polarons increases, by decreasing
K/t, the effective interaction between them changes from antiferromagnetic to ferromagnetic. The scaling to the thermodynamic limit suggests an attractive interaction in the strong-coupling regime
(Jh>t) and no binding in the weak limit ${(J}_{h}\ensuremath{\simeq}t).$
In this work, we demonstrate that accurate ground-state wave functions may be constructed for polarons in a fully ab initio setting across the wide range of couplings associated with both the large and small polaron limits. We present a single general unitary transformation approach which encompasses an ab initio version of the Lee-Low-Pines theory at weak coupling and the coherent-state Landau-Pekar framework at strong coupling, while interpolating between these limits in general cases. We show that perturbation theory around these limits may be performed in a facile manner to assess the accuracy of the approach, as well as provide an independent route to the ab initio properties of polarons. We test these ideas on the case of LiF, where the electron polaron is expected to be large and relatively weakly coupled, while the hole polaron is expected to be a strongly coupled small polaron.