Krylov Complexity in Supersymmetric Large-$N$ Quantum Mechanics

We study Krylov complexity in the large-$N$ planar limit of the supersymmetric matrix quantum mechanical Veneziano--Wosiek model. In particular, we discuss the special features emerging at the 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 Krylov complexity by numerical methods. We find that 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 in the bosonic sector a solvable model with $\mathfrak{sl}(2, \mathbb{R})$ symmetry which is a rank-1 modification of the Veneziano--Wosiek Hamiltonian, finding that it reproduces the previous features of complexity. We also introduce supercharges and extend the solvable model to the fermionic sector where we also compute analytically the Krylov complexity. 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 both in the original and in the solvable model. 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.