Multifractals Competing with Solitons on Fibonacci Optical Lattice

TL;DR

This paper investigates the stationary states of the nonlinear Schrödinger equation on a Fibonacci optical lattice, revealing a phase diagram with critical states, solitons, and a forbidden region, and showing the robustness of critical states against nonlinearity.

Contribution

It provides the first detailed phase diagram of energy spectra for nonlinear Schrödinger equations on Fibonacci lattices, highlighting the coexistence and robustness of critical states and solitons.

Findings

Critical states' spectrum remains unchanged with nonlinearity.

Identified three spectral regions: forbidden, critical, and solitonic.

Critical states are robust in the presence of stationary solitons.

Abstract

We study the stationary states for the nonlinear Schr\"odinger equation on the Fibonacci lattice which is expected to be realized by Bose-Einstein condensates loaded into an optical lattice. When the model does not have a nonlinear term, the wavefunctions and the spectrum are known to show fractal structures. Such wavefunctions are called critical. We present a phase diagram of the energy spectrum for varying the nonlinearity. It consists of three portions, a forbidden region, the spectrum of critical states, and the spectrum of stationary solitons. We show that the energy spectrum of critical states remains intact irrespective of the nonlinearity in the sea of a large number of stationary solitons.