Moving acoustical-mode polarons

Abstract

A general class of acoustical-phonon mode induced-polaron systems is considered. The electron-phonon interaction for these systems is described by a Fr\"ohlich-interaction Hamiltonian involving Fourier components proportional to ${q}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ is an arbitrary real number. The energy versus total momentum ($E\ensuremath{-}P$) relations for these polarons are investigated. At weak coupling, in the context of the intermediate coupling theory, it is found that when $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}\frac{1}{2}$ a linear structure is present in the $E\ensuremath{-}P$ relations. As $\ensuremath{\gamma}$ decreases from -1/2 to -1, this linear structure becomes apparent only for larger and larger values of $P$. For $\ensuremath{\gamma}>\ensuremath{-}\frac{1}{2}$, this linear structure is not present, but rather the $E\ensuremath{-}P$ relations cut off at a critical value of $P$. When $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}0$, the polaron mass for these systems grows large as the polaron velocity $v$ approaches the speed of sound $s$, but for $\ensuremath{\gamma}>0$ the polaron mass has a finite value as $v\ensuremath{\rightarrow}s$. As a function of $\ensuremath{\gamma}$, the self-energy reaches a maximum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.73$ and the $v\ensuremath{\rightarrow}0$ polaron mass reaches a minimum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.25$, independent of coupling strength. From a strong-coupling theory appropriate for moving polarons it is found that a linear structure in the $E\ensuremath{-}P$ relations is obtained for all $\ensuremath{\gamma}>\ensuremath{-}1$. At both weak and strong coupling it is unclear that the $E\ensuremath{-}P$ relations are meaningful when $\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}1$.