A bstract The Sachdev-Ye-Kitaev (SYK) model is a system of N Majorana fermions with random interactions and strongly chaotic dynamics, which at low energy admits a holographically dual description as two-dimensional Jackiw-Teitelboim gravity. Hence the SYK model provides a toy model of quantum gravity that might be feasible to simulate with near-term quantum hardware. Motivated by the goal of reducing the resources needed for such a simulation, we study a sparsified version of the SYK model, in which interaction terms are deleted with probability 1 −p . Specifically, we compute numerically the spectral form factor (SFF, the Fourier transform of the Hamiltonian’s eigenvalue pair correlation function) and the nearest-neighbor eigenvalue gap ratio r (characterizing the distribution of gaps between consecutive eigenvalues). We find that when p is greater than a transition value p 1 , which scales as 1/ N 3 , both the SFF and r match the values attained by the full unsparsified model and with expectations from random matrix theory (RMT). But for p < p 1 , deviations from unsparsified SYK and RMT occur, indicating a breakdown of holography in the highly sparsified regime. Below an even smaller value p 2 , which also scales as 1/ N 3 , even the spacing of consecutive eigenvalues differs from RMT values, signaling a complete breakdown of spectral rigidity. Our results cast doubt on the holographic interpretation of very highly sparsified SYK models obtained via machine learning using teleportation infidelity as a loss function.
The realization of holographic duality on near-term intermediate-scale quantum (NISQ) hardware requires identifying sparse Hamiltonians that retain essential gravitational features. While previous approaches rely on machine-learning optimization to find such models, we demonstrate that simple magnitude-based pruning of the Sachdev-Ye-Kitaev (SYK) model preserves holographic transport at sparsities ≈5%. By mapping the phase diagram of sparsified SYK (N ≤ 12), we identify a distinct phase boundary where quantum chaos, diagnosed via the Spectral Form Factor (SFF), dissolves, even while traversable wormhole signatures persist. This decoupling suggests that information scrambling can occur without the emergent geometry required for gravity. We validate the gravitational phenomenon of the chaotic phase by observing a quantized Shapiro time delay in the teleportation signal with perturbation, consistent with the emergence of bulk geometry. Finally, we show that this robustness is unique to all-to-all connectivity. Imposing linear local geometry destroys the holographic phase, highlighting non-locality as critical for quantum gravity simulations.
Recently, the physics of many-body quantum chaotic systems close to their ground states has come under intensified scrutiny. Such studies are motivated by the emergence of model systems exhibiting chaotic fluctuations throughout the entire spectrum [the Sachdev-Ye-Kitaev (SYK) model being a renowned representative] as well as by the physics of holographic principles, which likewise unfold close to ground states. Interpreting the edge of the spectrum as a quantum critical point, here we combine a wide range of analytical and numerical methods to the identification and comprehensive description of two different universality classes: the near edge physics of “sparse” and the near edge of “dense” chaotic systems. The distinction lies in the ratio between the number of a system's random parameters and its Hilbert space dimension, which is exponentially small or algebraically small in the sparse and dense case, respectively. Notable representatives of the two classes are generic chaotic many-body models (sparse) and invariant random matrix ensembles or chaotic gravitational systems (dense). While the two families share identical spectral correlations at energy scales comparable to the level spacing, the density of states and its fluctuations near the edge are different. Considering the SYK model as a representative of the sparse class, we apply a combination of field theory and exact diagonalization to a detailed discussion of its edge spectrum. Conversely, Jackiw-Teitelboim gravity is our reference model for the dense class, where an analysis of the gravitational path integral and random matrix theory reveal universal differences to the sparse class, whose implications for the construction of holographic principles we discuss. Published by the American Physical Society 2024
Quantum chaos is one of the distinctive features of the Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions in 0+1 dimensions with infinite-range two-body interactions, which is attracting a lot of interest as a toy model for holography. Here we show analytically and numerically that a generalized SYK model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic and retains most of its holographic features for a fixed value of the perturbation and sufficiently high temperature. However, a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. We speculate about the gravity dual of this transition.
We study a sparse Sachdev-Ye-Kitaev (SYK) model with
N Majoranas where only $\ensuremath{\sim}kN$ independent matrix elements are nonzero. We identify a minimum $k\ensuremath{\gtrsim}1$ 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 nonzero 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 $k\ensuremath{\rightarrow}1$ 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 tenfold 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
2m-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
28-fold degenerate spectra for
N=26.
The Sachdev-Ye-Kitaev (SYK) model is a cornerstone in the study of quantum chaos and holographic quantum matter. Real-world implementations, however, deviate from the idealized all-to-all connectivity, raising questions about the robustness of its chaotic properties. In this work, we investigate a quadratic SYK model with distance-dependent interactions governed by a power-law decay. By analytically and numerically studying the spectral form factor (SFF), we uncover how transitions present in the single-particle limit carry over to the many-body system. Nontrivial cancellations in the one-loop contributions lead to a robustness of the SFF under a considerable reduction of the interaction range. Further suppression leads to a breakdown of perturbation theory around the infinite-range path-integral saddle and the appearance of spectral regimes, marked by a higher dip and the emergence of a secondary plateau. Our results highlight the interplay between single-particle criticality and many-body dynamics, offering insights into the quantum chaos-to-localization transition and its reflection in spectral statistics.
The sparse version of the Sachdev-Ye-Kitaev (SYK) model reproduces essential features of the original SYK model while reducing the number of disorder parameters. In this Research Letter, we propose a further simplification of the model which we call the binary-coupling sparse SYK model. We set the nonzero couplings to be $\ifmmode\pm\else\textpm\fi{}1$, rather than being sampled from a continuous distribution such as Gaussian. Remarkably, this simplification turns out to be an improvement: The binary-coupling model exhibits strong correlations in the spectrum, which is the important feature of the original SYK model that leads to the quick onset of the random-matrix universality, more efficiently in terms of the number of nonzero terms. This model is better suited for analog or digital quantum simulations of quantum chaotic behavior and holographic metals due to its simplicity and scaling properties.