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arXiv:2401.07325 · Advanced Quantum Technologies

Randomly Pruning the Sachdev‐Ye‐Kitaev Model

Richard Berkovits

Published Jan 25, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract The Sachdev‐Ye‐Kitaev model (SYK) is renowned for its short‐time chaotic behavior, which plays a fundamental role in its application to various fields such as quantum gravity and holography. The Thouless energy, representing the energy scale at which the universal chaotic behavior in the energy spectrum ceases, can be determined from the spectrum itself. When simulating the SYK model on classical or quantum computers, it is advantageous to minimize the number of terms in the Hamiltonian by randomly pruning the couplings. In this paper, it is demonstrated that even with a significant pruning, eliminating a large number of couplings, the chaotic behavior persists up to short time scales. This is true even when only a fraction of the original couplings in the fully connected SYK model, specifically , is retained. Here, represents the number of sites, and . The properties of the long‐range energy scales, corresponding to short time scales, are verified through numerical singular value decomposition (SVD) and level number variance calculations.

DOI
10.1002/qute.202300222

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Abstract The Sachdev‐Ye‐Kitaev model (SYK) is renowned for its short‐time chaotic behavior, which plays a fundamental role in its application to various fields such as quantum gravity and holography. The Thouless energy, representing the energy scale at which the universal chaotic behavior in the energy spectrum ceases, can be determined from the spectrum itself. When simulating the SYK model on classical or quantum computers, it is advantageous to minimize the number of terms in the Hamiltonian by randomly pruning the couplings. In this paper, it is demonstrated that even with a significant pruning, eliminating a large number of couplings, the chaotic behavior persists up to short time scales. This is true even when only a fraction of the original couplings in the fully connected SYK model, specifically , is retained. Here, represents the number of sites, and . The properties of the long‐range energy scales, corresponding to short time scales, are verified through numerical singular value decomposition (SVD) and level number variance calculations.
The short time (large energy) behavior of the Sachdev-Ye-Kitaev model (SYK) is one of the main reasons for the growing interest garnered by this model. True chaotic behavior sets in at the Thouless time, which can be extracted from the energy spectrum. In order to do so, it is necessary to unfold the spectrum, i.e., to filter out global tendencies. Using a simple ensemble average for unfolding results in a parametically low estimation of the Thouless energy. By examining the behavior of the spectrum as the distribution of the matrix elements is changed into a log-normal distribution, it is shown that the sample-to-sample level spacing variance determines this estimation of the Thouless energy. Using the singular value decomposition method, which filters out these global sample-to-sample fluctuations, the Thouless energy becomes parametrically much larger, essentially of the order of the band width. It is shown that the SYK model is non-self-averaging even in the thermodynamic limit which must be taken into account in considering its short time properties.
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.
We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the kkk-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth- and sixth-order energy cumulants vanish in the limit of a large number of particles $N\ensuremath{\rightarrow}\ensuremath{\infty}$, which is consistent with a Gaussian spectral density. However, for finite NNN, the tail of the average spectral density is well approximated by a semicircle law. The specific heat coefficient, determined numerically from the low-temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on NNN. For larger energy separations we identify an energy scale that grows with NNN, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity dual.
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.
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 version of the Sachdev-Ye-Kitaev (SYK) model defined on random hypergraphs constructed either by a random pruning procedure or by randomly sampling regular hypergraphs. The resulting model has a new parameter, kkk, defined as the ratio of the number of terms in the Hamiltonian to the number of degrees of freedom, with the sparse limit corresponding to the thermodynamic limit at fixed kkk. We argue that this sparse SYK model recovers the interesting global physics of ordinary SYK even when kkk is of order unity. In particular, at low temperature the model exhibits a gravitational sector which is maximally chaotic. Our argument proceeds by constructing a path integral for the sparse model which reproduces the conventional SYK path integral plus gapped fluctuations. The sparsity of the model permits larger scale numerical calculations than previously possible, the results of which are consistent with the path integral analysis. Additionally, we show that the sparsity of the model considerably reduces the cost of quantum simulation algorithms. This makes the sparse SYK model the most efficient currently known route to simulate a holographic model of quantum gravity. We also define and study a sparse supersymmetric SYK model, with similar conclusions to the non-supersymmetric case. Looking forward, we argue that the class of models considered here constitute an interesting and relatively unexplored sparse frontier in quantum many-body physics.