Gaussian Ensemble of varying randomness and Sparse Sachdev-Ye-Kitaev model

We study a system of N qubits, at large N, 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. Starting with the GUE, we study the resulting behavior 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 behavior changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N.