Dual polyhedron

Quantum gravity

Combinatorics

Physics

Quantum dynamics

Mathematics

Mathematical physics

Random matrix

Statistical physics

Quantum chaos

Eigenvalues and eigenvectors

Fibonacci number

Geometry

Quantum mechanics

Degenerate energy levels

Quantum

Scaling

Physical review. D/Physical review. D.

We study a sparse Sachdev-Ye-Kitaev (SYK) model with $N$ Majoranas where only $\ensuremath{\sim}kN$ independent matrix elements are nonzero. We identify a minimum $k\ensuremath{\gtrsim}1$ 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 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 $k\ensuremath{\rightarrow}1$ 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 $N=26$.

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