Reentrant localization phenomenon in one-dimensional cross-stitch lattice with flat band

Abstract

In this work, we numerically study the localization properties in a quasi-periodically modulated one-dimensional cross-stitch lattice with a flat band. When <inline-formula><tex-math id="M3">\begin{document}$\varDelta\neq0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231393_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231393_M3.png"/></alternatives></inline-formula>, it is found that there are two different quasi-periodic modulation frequencies in the system after the local transformation, and the competing modulation by two frequencies may lead to the reentrant localization transition in the system. By numerically solving the fractal dimension, the average inverse participation ratio, and the average normalized participation ratio, we confirm that the system can undergo twice localization transitions. It means that the system first becomes localized as the disorder increases, at some critical points, some of the localized states go back to the delocalized ones, and as the disorder further increases, the system again becomes fully localized. By the scalar analysis of the normalized participation ratio, we confirm that reentrant localization stably exists in the system. And the local phase diagram is also obtained. From the local phase diagram, we find that when <inline-formula><tex-math id="M4">\begin{document}$1.6<\varDelta<1.9$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231393_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231393_M4.png"/></alternatives></inline-formula>, the system undergoes a cascade of delocalization-localization-delocalization-localization transition by increasing <i>λ</i>. When <inline-formula><tex-math id="M5">\begin{document}$\varDelta=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231393_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231393_M5.png"/></alternatives></inline-formula>, there exists only one quasi-periodic modulation frequency in the system. And we analytically obtain the expressions of the mobility edges, which are in consistence with the numerical studies by calculating the fractal dimension. And the system exhibits one localization transition. This work could expand the understanding of the reentrant localization in a flat band system and offers a new perspective on the research of the reentrant localization transition.