Quantum Chaos and Holographic Tensor Models

TL;DR

This paper analyzes the spectral properties of a simple holographic tensor model, revealing features of quantum chaos and random matrix behavior similar to SYK, with unique degeneracies and symmetries that influence its classification.

Contribution

The paper explicitly diagonalizes a Gurau-Witten tensor model, identifying its spectral features, degeneracies, and symmetry properties, and classifies its chaos behavior within the Altland-Zirnbauer framework.

Findings

Spectral form factor resembles SYK after running time average

Spectrum shows large degeneracies of intermediate states

Level spacing suggests quantum chaos despite degeneracies

Abstract

A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-$N$ behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, after a running time average, the spectral form factor exhibits striking qualitative similarities to SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, the spectrum has gaps and is not (linearly) rigid. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.