Sparse Sachdev-Ye-Kitaev model, quantum chaos, and gravity duals

We study a sparse Sachdev-Ye-Kitaev (SYK) model with N Majoranas where only ∼kN independent matrix elements are nonzero. We identify a minimum k 1 for quantum chaos to occur by a level statistics analysis. The spectral density in this region, and for a larger k, is still given by the Schwarzian prediction of the dense SYK model, though with renormalized parameters. Similar results are obtained for a beyond linear scaling with N of the number of nonzero matrix elements. This is a strong indication that this is the minimum connectivity for the sparse SYK model to still have a quantum gravity dual. We also find an intriguing exact relation between the leading correction to moments of the spectral density due to sparsity and the leading 1/d correction of Parisi's U(1) lattice gauge theory in a d-dimensional hypercube. In the k→1 limit, different disorder realizations of the sparse SYK model show emergent random matrix statistics that for fixed N can be in any universality class of the tenfold way. The agreement with random matrix statistics is restricted to short-range correlations, no more than a few level spacings, in particular in the tail of the spectrum. In addition, emergent discrete global symmetries in most of the disorder realizations for k slightly below one give rise to 2m-fold degenerate spectra, with m being a positive integer. For k=3/4, we observe a large number of such emergent global symmetries with a maximum 28-fold degenerate spectra for N=26.