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

We study a sparse Sachdev-Ye-Kitaev (SYK) model with $N$ Majoranas where only $\sim k N$ independent matrix elements are non-zero. We identify a minimum $k \gtrsim 1$ 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 non-zero 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 $k \to 1$ 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 ten-fold 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 $N = 26$.