Sachdev-Ye-Kitaev model: Non-self-averaging properties of the energy spectrum

The short time (large energy) behavior of the Sachdev-Ye-Kitaev model (SYK) is one of the main motivation to the growing interest garnered by this model. True chaotic behaviour sets in at the Thouless time, which can be extracted from the energy spectrum. In order to do so, it is necessary to unfold the spectrum, i.e., to filter out global tendencies. Using a simple ensemble average for unfolding results in a parametically low estimation of the Thouless energy. By examining the behavior of the spectrum as the distribution of the matrix elements is changed into a log-normal distribution it is shown that the sample to sample level spacing variance determines this estimation of the Thouless energy. Using the singular value decomposition method, SVD, which filters out these sample to sample fluctuations, the Thouless energy becomes parametrically much larger, essentially of order of the band width. It is shown that the SYK model in non-self-averaging even in the thermodynamic limit which must be taken into account in considering its short time properties.