K. Yonemitsu, TIME-DEPENDENT MEAN-FIELD THEORY FOR TUNNELING IN ELECTRON-PHONON SYSTEMS, Physical review. B, Condensed matter, 50(5), 1994, pp. 2899-2920
A semiclassical method is presented for tunneling of self-trapped stat
es in many-body systems with electrons and phonons. An overcomplete se
t of Slater determinants, lattice coordinates, and lattice momenta is
used to represent a functional integral. Stationary phase equations ar
e solved numerically without any constraint on the dynamics of electro
ns and phonons, i.e., without the use of the adiabatic approximation.
To evaluate transition amplitudes, we integrate over small fluctuation
s in both electronic and phonon degrees of freedom, keeping their time
order correctly. This method can be applied to general electron-phono
n systems and is useful when self-trapped states have complex structur
es in charge or spin densities and lattice displacements. The effectiv
e hopping strength is calculated for a self-trapped kink in the commen
surate charge-density-wave state in one dimension. At strong coupling
in the Holstein and attractive Hubbard models, where tunneling involve
s effectively a single tightly bound bipolaron, this method reproduces
previous analytic results. New results are obtained at intermediate c
oupling where the kink is extended over a couple of lattice sites and
for models with both electron-electron and electron-phonon interaction
s.