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 reasons for the growing interest garnered by this model. True chaotic behavior 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, which filters out these global sample-to-sample fluctuations, the Thouless energy becomes parametrically much larger, essentially of the order of the band width. It is shown that the SYK model is non-self-averaging even in the thermodynamic limit which must be taken into account in considering its short time properties.