Krylov Complexity in Supersymmetric Large-$N$ Quantum Mechanics

Abstract

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 $λ=1$. 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 $λ\neq1$ the Krylov complexity $K(t)$ exhibits oscillatory behavior, while at the critical coupling $λ=1$ 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 $ω$, with the free limit $ω\to 0$ corresponding to the critical $λ\to 1$ limit.

Figures (8)

• Figure 1: Convergence of the spectrum of the planar Hamiltonians $H^{F=0}$ and $H^{F=1}$ at $\lambda=1/2$. Supersymmetric pairing of levels is rather accurate in panel (b).
• Figure 2: Numerical analysis of bosonic complexity $K^{0}(t; \lambda)$. The left panel is for $\lambda=2/3$, a generic value smaller than 1. Blue, red, orange, purple, and black lines correspond to $\mathsf{K}=30,50,100,150,250$. Further increasing $\mathsf{K}$ does not significantly change the complexity in this temporal window. The final complexity oscillates. Similar behaviour is observed for $\lambda>1$. In the right panel, we show results for $\lambda=1$. Curves from bottom to top correspond to $\mathsf{K}=50,100,150,200,250$. At the critical coupling, we see that the complexity saturates at large times for any fixed $\mathsf{K}$, while the envelope of the curves grows as $\mathsf{K}$ increases and grows approximately as $t^{2}$ (the dashed blue line is $1.1\, t^{2}$).
• Figure 3: Numerical analysis of fermionic complexity $K^{1}(t; \lambda)$. The left panel is for $\lambda=2/3$, a generic value smaller than 1. Blue, red, orange, purple, and black lines correspond to $\mathsf{K}=30,50,100,200,250$. Further increasing $\mathsf{K}$ does not change complexity appreciably in this temporal window. The final complexity oscillates. Values $\lambda>1$ are similar. In the right panel, we show results for $\lambda=1$. Curves from bottom to top correspond to $\mathsf{K}=50,100,150,200,250$. At the critical coupling, we see that complexity saturates for any $\mathsf{K}$ and an enveloping curve emerges. It grows $\sim t^{2}$ (the dashed blue line is $3.3\, t^{2}$).
• Figure 4: Analysis of the sector $F=0$ for $M=1$. The curves in panel (a) for $\lambda=2/3$ correspond to $\mathsf{K}=50,100,150$, while the dashed line represents the exact result (\ref{['6.6']}). The curves in panel (b) for $\lambda=1$ correspond to $\mathsf{K}=50,100,150,200,250$, and the enveloping dashed line is the first expression in (\ref{['6.12']}), i.e. the quadratic function $2t^{2}$.
• Figure 5: Analysis of the sector $F=0$ for $M=2$. The curves in panel (a) for $\lambda=2/3$ correspond to $\mathsf{K}=50,100,150,200$, while the dashed line represents the exact result (\ref{['6.6']}). The curves in panel (b) for $\lambda=1$ correspond to $\mathsf{K}=50,100,150,200,250$, and the enveloping dashed line is the second expression in (\ref{['6.12']}), i.e. the quartic function $2t^{2}(1+3t^{2})$.