Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas

Abstract

We propose and test logarithmic Krylov (logK) complexity, an operator growth measure akin to Krylov complexity defined through a replica approach, as a viable probe of early-time operator scrambling without false positives. In finite-dimensional quantum systems, such as the Lipkin--Meshkov--Glick (LMG) model and the mixed-field Ising model at the chaotic point, we provide numerical evidence that logK-complexity discriminates between genuine and saddle-dominated scrambling at early times, correctly avoiding the exponential contribution coming from the unstable saddle in the former case, and closely tracking the conventional Krylov complexity in the latter. In integrable quantum systems admitting infinite-dimensional Krylov subspaces, such as the SYK$_{2}$ model and the quantum inverted harmonic oscillator, we show that by modifying the Krylov spreading operator, obtained through generalizing the analytic continuation procedure in the replica trick, the logK complexity can be refined to capture the integrable properties of the theories. We supplement these analyses by extending the Krylov formalism in classical dynamical systems and defining classical versions of these operator growth measures, showing that the false positives arising from unstable saddles in classical phase space are non-existent.

Figures (7)

• Figure 1: Comparison of $\log(K(t))$\ref{['eq:KrylovCFT']} (blue, dashed) and $\log(\mathbf{E}_{K}(t))$\ref{['eq:eLogKCFT']} (orange) for $\eta=\alpha=1$.
• Figure 2: Krylov complexity $K(t)$ and elogK complexity $\mathbf{E}_{K}(t)$ in the LMG model for initial operator $\hat{z}$ with classical exponent $\lambda_{\mathrm{saddle}}=\sqrt{3}$. The horizontal axis is scaled with the slope of $b_{n}$ by $\alpha$, whereas the vertical axis is scaled by the Krylov dimension $D_{\mathcal{K}}=1300$. Upper left panel:$K(t)$ (blue, dashed) and $\mathbf{E}_{K}(t)$ (orange) for $s=25$. The inset is the log plot where the red dashed line is of slope $\lambda_{\mathrm{saddle}}=\sqrt{3}$. Upper right panel: The corresponding Lanczos coefficients with linear fitting by $\alpha=0.8701$. Middle left panel:$K(t)$ for $s=25$. The blue dot-dashed data are from numerical calculation. The red dashed curve is the initial time fitting of the quadratic polynomial function at $0<t_{\mathrm{In}}\lesssim1$. The black curve is the exponential function fitting up to the scrambling time $t^*$. Middle right panel:$\mathbf{E}_{K}(t)$ for $s=25$. The orange dot-dashed data are from numerical calculation. The red dashed curve is the initial time fitting of the quartic polynomial function at $0<t_{\mathrm{In}}\lesssim1$. The black curve is the fitting up to the scrambling time $t^*$. Lower left panel: Comparison of $K(t)$ and $\mathbf{E}_{K}(t)$ for $s=25$ and $s=75$. Lower right panel: Late-time saturation of $K(t)$ and $\mathbf{E}_{K}(t)$ for $s=25$. For $s=25$, the least numerical precision of the Krylov basis is on the order of $10^{-600}$, whereas for $s=75$, it is on the order of $10^{-100}$.
• Figure 3: Upper left panel: Lanczos coefficients for the operator $\hat{S}_{5}^{z}+\hat{S}_{6}^{z}$ in the mixed-field Ising model with $h_{x}=-1.05$, $h_{z}=0.5$ with sites $L=10$ in the negative parity sector. The linear growth with slope $\alpha=0.8516$ indicates the exponential growth of Krylov complexity. Upper right panel:$K(t)$ (blue, dashed) and $\mathbf{E}_{K}(t)$ (orange) for the same parameters and operator in the negative parity sector. The horizontal axis is scaled with the slope of $b_{n}$ by $\alpha$, whereas the vertical axis is scaled by the Krylov dimension $D_{\mathcal{K}}=245521$. The inset is the log plot. Lower panel: Late-time saturation of $K(t)$ and $\mathbf{E}_{K}(t)$ in the negative parity sector with Krylov dimension $D_{\mathcal{K}}=3081$ for sites $L=7$ and operator $\hat{S}_{3}^{z}+\hat{S}_{5}^{z}$ with slope $\alpha=1.0848$. For $L=10$, the least numerical precision of the Krylov basis is on the order of $10^{-360}$, whereas for $L=7$, it is on the order of $10^{-820}$.
• Figure 4: Schematic figure for the phase space trajectory of the linearized dynamics around the unstable saddle point. The blue curve represents the solution \ref{['eq:UnstableSaddleSol']}. The orange and dashed region is the strip \ref{['eq:PhaseSpSaddleRegion']} within which the orbit of the initial phase space point $(\delta a^{+}_{0},\delta a^{-}_{0})$ evolves exponentially.
• Figure 5: Left panel: Normalized Krylov complexity $K\left(t\right)$ (\ref{['eq:ClassKrylovSaddleDomExq']}) (blue) and un-normalized Krylov complexity (\ref{['eq:ClassKrylovSaddleDomExqVolStrip']}) with strip constraint $12\tilde{K}\left(t\right)/\delta^2$ (orange) for initial function $q$. Right panel: Normalized Krylov complexity $K\left(t\right)$ (\ref{['eq: NormKq2']}) (blue) and elogK complexity (orange) $\mathbf{E}_K(t)$ from (\ref{['eq: NormLKq2']}) (orange) for initial function $q^2$. We chose $\lambda_{\text{cl}}=1$ for both plots.