Quantum Chaos and Holographic Tensor Models

TL;DR

This work explicitly diagonalizes the simplest nontrivial Gurau-Witten tensor model (D=3, n=2) and analyzes its spectral and late-time properties. It finds SYK-like features in the spectral form factor after running-time averaging, indicating chaotic dynamics, but also reveals a large mid-spectrum degeneracy and a spectral mirror symmetry. The degeneracies soften into level repulsion upon unfolding, pointing to random-matrix-type behavior controlled by the BDI (chiral orthogonal) class rather than Dyson ensembles. The results suggest holographic tensor models can exhibit quantum-chaotic behavior without quenched disorder, while highlighting unique spectral structures that differ from SYK and warrant further study at larger N and D.

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.

Figures (9)

• Figure 1: The MatrixPlot of Hamiltonian \ref{['Ham']}.
• Figure 2: The MatrixPlot of an SYK Hamiltonian for a single draw from the ensemble. We are considering the case $N=16$, with really real Gamma matrices.
• Figure 3: The density of states. The d.o.s is symmetric: the slight asymmetry is an artifact of the binning of the eigenvalues.
• Figure 4: The integrated density of states. The jump around zero is a result of the degeneracy at $E=0$.
• Figure 5: The SFF for $\beta=0$