Probing the metallic energy spectrum beyond the Thouless energy scale using singular value decomposition

Disordered quantum systems feature an energy scale known as the Thouless energy. For ranges below this scale, properties of spectrum can be described by random matrix theory. Above a different behavior sets in. metallic system it was shown long ago Altshuler and Shklovskii [Sov. Phys. JETP 64, 127 (1986)] that number variance should increase power law with dependent on only dimensionality system. Although tantalizing hints at were seen in previous numerical studies, is quite difficult to verify prediction using standard local unfolding methods. Here we use method, i.e., singular value decomposition, establish connection between scree plot (the values ranked their amplitude) variance. Thus, are able numerically Shklovskii's for disordered three-, four-, five-dimensional single-electron Anderson models square lattices regime. The same method could applied such Sachdev-Ye-Kitaev model various interacting many-body which localization occurs. It recently reported exhibit energy, analyzing spectrum's larger scales much current interest.