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

In this work, we numerically study the localization properties in a quasi-periodically modulated one-dimensional cross-stitch lattice with a flat band. When , it is found that there are two different quasi-periodic modulation frequencies in the system after the local transformation, and the competing modulation bytwo frequencies may lead to the reentrant localization transition in the system. By numerically solving thefractal dimension, the average inverse participation ratio, and the average normalized participation ratio, weconfirm that the system can undergo twice localization transitions. It means that the system first becomeslocalized as the disorder increases, at some critical points, some of the localized states go back to the delocalizedones, and as the disorder further increases, the system again becomes fully localized. By the scalar analysis ofthe normalized participation ratio, we confirm that reentrant localization stably exists in the system. And thelocal phase diagram is also obtained. From the local phase diagram, we find that when , thesystem undergoes a cascade of delocalization-localization-delocalization-localization transition by increasing lambda.When , there exists only one quasi-periodic modulation frequency in the system. And we analyticallyobtain the expressions of the mobility edges, which are in consistence with the numerical studies by calculatingthe fractal dimension. And the system exhibits one localization transition. This work could expand theunderstanding of the reentrant localization in a flat band system and offers a new perspective on the research ofthe reentrant localization transition