August 21, 2025Open Access

Sparsity dependence of Krylov state complexity in the SYK model

RJRaghav G. JhaNorth Carolina State University RRRanadeep RoyThe Ohio State University

Key Points

Krylov state complexity remains stable as sparsification increases, suggesting critical thresholds for holography.
The peak complexity does not change for sparse parameters above kmin, indicating complex interactions in majorana fermions.
Using Krylov complexity analysis, we explored correlations with holographic features in SYK-type models.
Findings may clarify critical thresholds (kc) in SYK models, potentially impacting further investigations in quantum gravity.

Abstract

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.

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