Quantum chaos and holographic tensor models

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, the spectral form factor exhibits a dip-rampplateau structure after a running time average, in qualitative agreement with 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, there are gaps in the spectrum. 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.

Towards a finite-N hologram

“…The explicit case diagonalized in [31] and found to have some chaos-like behavior was an odd n case, the higher even n cases were inaccessible to numerical diaogonalization. We find the scenario that the odd and even cases are qualitatively different to be quite unlikely (note also that in the colored model, the even case already exhibited chaos [30]), afterall it seems reasonable to expect that the large-N limit of the even and odd cases should match. But we are not sure if this is a watertight way to rule out this malicious possibility, which is why we have decided to menion it here.…”

“…In this paper, we will argue that the holographic tensor models that have recently emerged [26][27][28][29][30][31][32][33][34][35] in the context of SYK models are a potential candidate for such fully solvable theories. See [76][77][78][79][80] for other related works on tensor models.…”

“…Indeed, in [30,31] these theories were numerically diagonalized for small (and fixed) values of some appropriately chosen N . But it was also observed that as the N was increased, the matrix sizes grew exponentially as ∼ exp(N # ) where # is ≥ 1, and depends on the specific model.…”

“…In particular, constructing singlets in order to make contact with a "closed string" bulk perspective is much more intuitive when one has at least some analytic understanding. Furthermore, even in the simplest cases considered in [30,31] explicit diagonalization on a computer was too demanding for the eigenvectors to be found: only the eigenvalues were found in [30,31].…”

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

“…As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.…”