Electron model on Truchet tiling: Extended-to-localized transitions and asymmetric spectrum

Motivated by recent advances in the realization of Truchet-tiling structures in molecular networks and metal-organic frameworks, we investigate the wave localization issue in this kind of structure. We introduce an electron model based on random Truchet tilings, square lattices with randomly oriented diagonal links, and uncover a rich interplay between spectral and localization phenomena. By varying the strength of diagonal couplings, we explore successive transitions from an extended phase, through a regime with a mobility edge, to a fully localized phase. The energy-resolved fractal dimension analysis captures the emergence and disappearance of mobility edges, while an anomalous shift and asymmetry in the Van Hove singularity are identified as key signatures of the underlying disordered Truchet-tiling structure. Notably, by using the finite-size scaling of level spacing statistics, we clarify that the transition occurs at a finite level of disorder even in the two-dimensional system. It turns out that the suppression of long-range fluctuation in the random Truchet tiling, known as the hyperuniformity, gives rise to such an unconventional extended-to-localized transition. Our findings position Truchet-tiled electron systems as a versatile platform for engineering disorder-driven localization and interaction effects in amorphous quantum materials and photonic architectures.