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