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

We study a sparse Sachdev-Ye-Kitaev (SYK) model with N Majoranas where only similar to kN independent matrix elements are nonzero. We identify a minimum k greater than or similar to 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 2(m)-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 2(8)-fold degenerate spectra for N = 26.