Krylov Complexity in Supersymmetric Large-$N$ Quantum Mechanics

Krylov complexity has recently emerged as a useful probe of operator growth and quantum dynamics in many-body systems and holographic dualities. In this paper we study its behavior in the Veneziano--Wosiek model, a supersymmetric matrix quantum mechanical model admitting a large-$N$ planar limit with manifest weak-strong duality and a critical transition at the 't Hooft coupling $\lambda=1$. Starting from selected states in the sectors with fermion number 0 and 1, related by supersymmetry, we analyze the time dependence of the numerical complexity. For $\lambda\neq1$ the Krylov complexity $K(t)$ exhibits oscillatory behavior, while at the critical coupling $\lambda=1$ it grows quadratically in time, $K(t)\sim t^2$, with sector-dependent amplitudes. To obtain analytical insight, we study a companion model defined by a rank-1 modification of the Veneziano--Wosiek Hamiltonian, which admits explicit supercharges. In this model the Krylov complexity can be computed exactly and reproduces the behavior observed in the original model. Higher degree-$M$ Krylov complexities, defined as expectation values of powers of Lanczos index, are also computed and grow polynomially in time $\sim t^{2M}$ at the critical point in both models. This behavior is closely analogous to the spreading of a localized squeezed state in a one-dimensional quantum harmonic oscillator of frequency $\omega$, with the free limit $\omega\to 0$ corresponding to the critical $\lambda\to 1$ limit.