Asymptotic dependence of Gross–Tulub polaron ground-state energy in the strong coupling region - Kashirina, N.I. (Semiconductor Physics Quantum Electronics & Optoelectronics, 2017)
The properties of the translationally invariant polaron functional have been investigated in the strong- and extremely-strong-coupling regimes. It has been shown that the Gross–Tulub polaron functional obtained earlier using field-theoretic methods was derived only for the region α ≤ 10, where α is the Fröhlich constant of the electron-phonon coupling. Various representations of exact and approximate polaron functionals have been considered. Asymptotic dependences of the polaron energy have been obtained using a functional extending the Gross–Tulub functional to the region of extremely strong coupling. The asymptotic dependence of polaron energies for an extremely strong coupling is Ep ≈ −0.31683α⁴/ ³ (for the one-parameter variational function fₖ), and Ep ≈ −0.31767α⁴/ ³ (for a two-parameter function f′ₖ ). It has been shown that the virial theorem 1:3:4 holds for the two-parameter function f′ₖ. Minimization of the approximate functional obtained by expanding the exact Gross–Tulub functional in a series on 1/α leads to a quadratic dependence of the polaron energy. This approximation is justified for α ≈ 7.5...8 . For a two-parameter function f′ₖ, the corresponding dependence has the form Ep ≈ −0.125α². However, the use of approximate functionals, in contrast to the strict variational procedure, when the exact polaron functional varies, does not guarantee obtaining the upper limit for the polaron energy.