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

TL;DR

This paper investigates a sparse version of the SYK model, identifying the minimal connectivity for quantum chaos and potential gravity duals, and explores spectral properties and emergent symmetries in the sparse regime.

Contribution

It introduces a sparse SYK model with minimal connectivity for quantum chaos and links spectral corrections to lattice gauge theory, revealing emergent symmetries and universality classes.

Findings

Quantum chaos occurs for a minimum connectivity $k \,\gtrsim\ 1$.

Spectral density matches Schwarzian predictions with renormalized parameters.

Emergent global symmetries lead to degenerate spectra and universality class variations.

Abstract

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 $N = 26$.