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.

💡 Research Summary

This paper investigates how breaking supersymmetry (SUSY) influences Krylov complexity in the $\mathcal{N}=2$ Sachdev‑Ye‑Kitaev (SYK) model. Two distinct deformations of the Hamiltonian are considered. The first is an irrelevant UV deformation that reduces the supersymmetry from $\mathcal{N}=2$ to $\mathcal{N}=1$, thereby removing the BPS ground‑state degeneracy while leaving the low‑energy (IR) physics essentially unchanged. The second is a mass deformation that adds a free two‑body SYK term, completely breaking supersymmetry ($\mathcal{N}=2\to0$) and lifting all degeneracies, pushing the system toward integrability.

Using Krylov subspace techniques, the authors construct the Krylov basis for a chosen initial operator (typically a fermionic creation or annihilation operator) by repeatedly applying the Liouvillian $L=