Pekar's ansatz and the strong coupling problem in polaron theory

Abstract

A detailed consideration is given to the translation-invariant theory of Tulub polaron constructed without the use of Pekar ansatz. A fundamental result of the theory is that the value of the polaron energy is lower than that obtained on the basis of Pekar ansatz which was considered as an asymptotically exact solution in the strong coupling limit. In the case of bipolarons the theory yields the best values of the coupling energy and critical parameters of their stability. Numerous physical consequences of the existence of translation-invariant polarons and bipolarons are discussed.

A translation invariant bipolaron in the Holstein model and superconductivity

“…According to the results obtained in Tulub (1962) and Lakhno (2015b), the spectrum of excited states of Hamiltonian (18) and (19) is determined by the expression:where are operators in which quadric form (19) is diagonal. Operators can be considered as operators of birth and annihilation of TI-bipolarons in excited states obeying Bose commutation relations:Renormalized frequencies involved in (34), according to Tulub (1962) and Lakhno (2015b), are determined by the equation for s :solutions of which give the spectrum of solutions.…”

“…Operators can be considered as operators of birth and annihilation of TI-bipolarons in excited states obeying Bose commutation relations:Renormalized frequencies involved in (34), according to Tulub (1962) and Lakhno (2015b), are determined by the equation for s :solutions of which give the spectrum of solutions.…”

“…As is shown in Lakhno (2015b), presentation of the wave function as a product of the electron wave function by the lattice one (Pekar ansatz) does not give an exact solution of Hamiltonian (1). A similar conclusion is valid for Hamiltonian (52).…”

“…It should also be noted that apart from the condition , for continuum TI-bipolarons to exist, the condition of continuity should also be met. According to Lakhno (2015b) it implies that the characteristic phonon vectors making the main contribution into the energy of TI-bipolarons should satisfy the inequality . From (27) it follows that the main contribution into the energy is given by the values of .…”

Asymptotic dependence of Gross–Tulub polaron ground-state energy in the strong coupling region

“…As shown in [11,12], the Tulub polaron functional [8] can be used to find the functional of the translation-invariant bipolaron. The discussion material devoted to the consideration of the asymptotic dependences of the polaron and bipolaron energy, which were found by the Tulub method [8], is described in the works [12][13][14][15][16].…”