ar X iv : c on d - m at / 9 91 02 79 v 1 1 8 O ct 1 99 9 Universal Selftrapping in Nonlinear Tight - binding Lattices

We show that nonlinear tight-binding lattices of different geometries and dimensionalities, display an universal selftrapping behavior. First, we consider the single nonlinear impurity problem in various tight-binding lattices, and use the Green's function formalism for an exact calculation of the minimum nonlinearity strength to form a stationary bound state. For all lattices, we find that this critical nonlinearity parameter (scaled by the energy of the bound state), in terms of the nonlinearity exponent, falls inside a narrow band, which converges to e 1/2 at large exponent values. Then, we use the Discrete Nonlinear Schrödinger (DNLS) equation to examine the selftrapping dynamics of a single excitation, initially localized on the single nonlinear site, and compute the critical nonlinearity parameter for abrupt dynamical selftrapping. For a given nonlinearity exponent, this critical nonlinearity, properly scaled, is found to be nearly the same for all lattices. Same results are obtained when generalizing to completely nonlinear lattices, suggesting an underlying selftrapping universality behavior for all nonlinear (even disordered) tight-binding lattices described by DNLS. The Discrete Nonlinear Schrödinger (DNLS) equation is a paradigmatic equation describing among others, dynamics of polarons in deformable media[1], local modes in molecular systems[2] and power exchange among non-linear coherent couplers in nonlinear optics[3]. Its most striking feature is the possibility of " selftrapping " , that is, the clustering of vibrational energy or electronic probability or electromagnetic energy in a small region of space. In a condensed matter context, the DNLS equation has the form i d C n d t = ǫ n C n + V m ′ C m − χ n |C n | α C n (1) where C n is the probability amplitude of finding the electron (or excitation) on site n of a d-dimensional lattice, ǫ n is the on–site energy, V is the transfer matrix element, χ n is the nonlinearity parameter at site n and α is the nonlinearity exponent. The prime in the sum in (1) restricts the summation to nearest–neighbors only. In the conventional DNLS case, α = 2 and χ n is proportional to the square of the electron-phonon coupling at site n.[4] Considerable work has been carried out in recent years to understand the stationary and dynamical properties of Eq. (1) in various cases. In particular, we point out the studies on the stability of the stationary solutions in one and two dimensions for the homogeneous case (ǫ …