Quantum Chaos and Holographic Tensor Models (1612.06330v3)

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.

• The paper numerically analyzes a Gurau-Witten tensor model to simulate quantum black holes, offering a disorder-free approach similar to SYK.
• Spectral features match SYK's chaotic signatures, including the dip-ramp-plateau, but uniquely reveal significant degeneracy in intermediate energy states.
• The study identifies a spectral mirror symmetry and classifies the model's randomness in the BDI class, differing from the Dyson ensembles seen in SYK models.

Quantum Chaos and Holographic Tensor Models: A Scholarly Overview

The paper "Quantum Chaos and Holographic Tensor Models" explores the fascinating intersection of quantum mechanics, holography, and random matrix theory. It investigates a novel approach to modeling quantum black holes through tensor models that emulate the Sachdev-Ye-Kitaev (SYK) model's large-$N$ behavior without necessitating the ensemble averaging typical of systems with quenched disorder. This foundational work by Krishnan et al. presents a rigorous exploration of the spectral properties of a Gurau-Witten tensor model, unraveling intriguing patterns that suggest a rich underlying chaotic structure.

Key Findings:

1. Model Selection and Numerical Analysis:
   • The study centers around explicit diagonalization of the simplest non-trivial Gurau-Witten tensor model. The tensor models discussed mirror the melonic large-$N$ behavior akin to the SYK model, offering an intriguing formulation for quantum black holes without relying on disorder averaging.
   • Utilizing real symmetric representations for $SO(32)$ gamma matrices, the researchers conquer computational limitations, handling the formidable challenges posed by large matrix sizes intrinsic to higher-dimensional models. The explicit Hamiltonian defined over a configuration of $N=32$ gamma matrices reveals sparse, intricate structural properties amenable to numerical analysis.
2. Spectral Features and Chaos Indicators:
   • The analysis surfaces spectral and late-time properties reminiscent of random matrix behavior — notably, the running time average of the spectral form factor displays striking parallels to SYK dynamics, manifesting features such as the dip-ramp-plateau structure indicative of chaos.
   • A significant departure from typical SYK models is discovered: the Gurau-Witten model exhibits a substantial degeneracy of intermediate energy states, paired with unique ground-state configurations. Such degeneracies, not intrinsic to SYK, showcase potential entropic signatures relevant to black hole state descriptions.
3. Symmetry and Ensemble Classification:
   • The spectral mirror symmetry observed stems from an underlying unitary operator anticommuting with the Hamiltonian. The paper meticulously traces this symmetry back to discrete transformations, identifying the model's randomness as governed by the BDI (chiral orthogonal) class rather than the Dyson ensembles seen in SYK.

Implications and Future Directions:

The study enriches our understanding of how holographic tensor models can be crafted as viable representations of quantum systems exhibiting chaotic behaviors. For theoretical physicists and researchers exploring holography, it opens pathways to consider tensor models not just as mathematical constructs but as frameworks capable of simulating the intricate dynamics of quantum black holes — expanding the horizons beyond conventional disorder-reliant models.

Moreover, the unique degeneracy and symmetry aspects revealed pose questions about how these features might translate to properties inherent to black holes, potentially offering insights into the quantum nature of gravitational systems and the universe's fundamental microstates. Going forward, it is vital to investigate larger $N$ behavior and generalize symmetry insights to unravel further the connections between tensor models and quantum gravity phenomena.

Conclusion:

Krishnan et al.'s research represents a pivotal step in broadening the scope of holographic models, providing an innovative lens to examine chaos in quantum mechanics through tensor constructs. The rigorous analysis, backed by strong numerical data, sets the stage for future endeavors in holography and quantum chaos, challenging pre-existing paradigms and inspiring speculative yet grounded approaches to understanding the quantum gravitational landscape.