Sparse random matrices and Gaussian ensembles with varying randomness

Abstract

We study a system of $N$ qubits with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed $q$ SYK theories. Our motivation is to understand the system at large $N$. In practice most of our calculations are carried out using exact diagonalisation techniques (up to $N=24$). Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in $N$. We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in $N$, and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile.

arXiv comment89 pages, 38 figures. v2: Section 3.1.1 added where we show that the rescaling can be modelled in terms of a weakening of the eigenvalue repulsion term. Appendix A is expanded

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• Sachdev-Ye-Kitaev model on a noisy quantum computer
• Probing quantum chaos through singular-value correlations in sparse non-Hermitian SYK model
• Two-local modifications of SYK model with quantum chaos
• Relaxation Fluctuations of Correlation Functions: Spin and Random Matrix Models
• Entanglement production in the Sachdev-Ye-Kitaev Model and its variants
• Many-body spectral transitions through the lens of the variable-range SYK2 model