Nonlinear impurity in a square lattice

We use the Green's-function formalism for an exact, numerical calculation o f the stationary states of an electron propagating in a square lattice in t he presence of a single, Holstein-type, impurity of arbitrary nonlinearity exponent. We find that two bound states exist above a certain exponent-depe ndent critical nonlinearity strength. The localization length of the lower (higher) energy bound state increases (decreases) with nonlinearity strengt h. The dynamics of an electron, initially placed on the impurity site, reve als a sharp, self-trapping transition for any nonzero nonlinearity exponent : below a certain nonlinearity threshold, the electron escapes from the imp urity site ballistically; above the threshold, there is partial trapping at the impurity site while the untrapped fraction escapes to infinity, also b allistically. The self-trapping features are sharper in time and space than for its one-dimensional analogue.