We study the Krylov state complexity of the Sachdev-Ye-Kitaev (SYK) model for N≤28 Majorana fermions with q-body fermion interaction with q=4, 6, 8 for a range of sparse parameter k that controls the number of remaining terms in the original SYK model after sparsification. The critical value of k below which the model ceases to be holographic, denoted kc, has been subject of several recent investigations. Using Krylov complexity as a probe, we find that the peak value of complexity does not change as we increase k beyond k≥kmin at large temperatures. We argue that this behavior is related to the change in the holographic nature of the Hamiltonian in the sparse SYK-type models such that the model is holographic for all k≥kmin≈kc. Our results provide a novel way to determine kc in SYK-type models.
Jha et al. (Thu,) studied this question.