Krylov Complexity of Supersymmetric SYK Models

We study the effect of supersymmetry breaking on Krylov complexity in the $\mathcal{N}=2$ SYK model under irrelevant and mass deformations of the Hamiltonian. The irrelevant deformation breaks $\mathcal{N}=2$ supersymmetry down to $\mathcal{N}=1$, while the mass deformation breaks supersymmetry completely. Using Krylov subspace methods, we analyze the Lanczos sequence, Krylov dimension, complexity, and entropy at finite system size as functions of deformation strength. Both deformations enlarge the Krylov space, but the mass deformation, which completely lifts the energy degeneracy, generates a stronger enhancement. Krylov complexity exhibits initial quadratic growth, followed by linear growth, across both deformations. We observe that both deformations increase the quadratic and linear growth rates of Krylov complexity at early times. At late times, the irrelevant deformation increases the saturation complexity as a fraction of the Krylov dimension, while the mass deformation decreases it. This reveals distinct signatures of how supersymmetry breaking impacts quantum complexity.