Krylov complexity and chaos in deformed SYK models

Krylov complexity has recently been proposed as a quantum probe of chaos. The Krylov exponent characterising the exponential growth of Krylov complexity is conjectured to upper-bound the Lyapunov exponent. We compute the Krylov and the Lyapunov exponents in the Sachdev-Ye-Kitaev model and in some of its deformations. We do this analysis both at infinite and finite temperatures, in models where the number of fermionic interactions is both finite and infinite. We consider deformations that interpolate between two regions of near-maximal chaos and deformations that become nearly-integrable at low temperatures. In all cases, we find that the Krylov exponent upper-bounds the Lyapunov one. However, we find that while the Lyapunov exponent can have non-monotonic behaviour as a function of temperature, in all studied examples the Krylov exponent behaves monotonically. For instance, we find models where the Lyapunov exponent goes to zero at low temperatures, while the Krylov exponent saturates to its maximal bound. We speculate on the possibility that this monotonicity might be a generic feature of the Krylov exponent in quantum systems evolving under unitary evolution.