The fractional saturable impurity

TL;DR

This paper investigates how fractional calculus affects impurity states in a 1D lattice, revealing significant deviations from standard models and demonstrating complete confinement at low fractional exponents.

Contribution

It introduces a fractional Laplacian approach to analyze impurity states, providing analytical and numerical results for bound states and transmission in fractional lattices.

Findings

Bound state energies depend on fractional exponent and impurity strength.

Transmission shows strong deviations from standard cases at small fractional exponents.

Complete confinement occurs as fractional exponent approaches zero.

Abstract

We examine analytically and numerically the effect of fractionality on a saturable bulk and surface impurity embedded in a 1D lattice. We use a fractional Laplacian introduced previously by us, and by the use of lattice Green functions we are able to obtain the bound state energies and amplitude profiles, as a function of the fractional exponent $s$ and saturable impurity strength $\chi$ for both, surface and bulk impurity. The transmission is obtained in closed form as a function of $s$ and $\chi$, showing strong deviations from the standard case, at small fractional exponent values. The selftrapping of an initially-localized excitation is qualitatively similar for the bulk and surface mode, but in all cases complete confinement is obtained at $s\rightarrow 0$, as shown theoretically and observed numerically.