Nonlinear Impurity in a Lattice: Dispersion Effects

TL;DR

This paper analyzes how second-nearest-neighbor hopping influences bound state formation in a one-dimensional nonlinear lattice, revealing that increased dispersion facilitates bound states at lower nonlinearity levels.

Contribution

It provides a closed-form solution for bound states in a nonlinear impurity model with extended hopping, highlighting the role of second-nearest-neighbor interactions in bound state formation.

Findings

Second-nearest-neighbor hopping lowers the nonlinearity threshold for bound states.

A phase diagram maps bound state existence as a function of nonlinearity and hopping ratios.

Enhanced dispersion enables earlier self-trapping transitions in the lattice.

Abstract

We examine the bound state(s) associated with a single cubic nonlinear impurity, in a one-dimensional tight-binding lattice, where hopping to first--and--second nearest neighbors is allowed. The model is solved in closed form {\em v\`{\i}a} the use of the appropriate lattice Green function and a phase diagram is obtained showing the number of bound states as a function of nonlinearity strength and the ratio of second to first nearest--neighbor hopping parameters. Surprisingly, a finite amount of hopping to second nearest neighbors helps the formation of a bound state at smaller (even vanishingly small) nonlinearity values. As a consequence, the selftrapping transition can also be tuned to occur at relatively small nonlinearity strength, by this increase in the lattice dispersion.