Universal relation for operator complexity

TL;DR

The paper investigates how Krylov complexity $C_K$ and operator entropy $S_K$ evolve during operator growth, uncovering a universal long-time relation $S_K\sim \log C_K$ for irreversible dynamics. Leveraging the recursion method and the Lanczos coefficients $\{b_n\}$, it connects short-time quadratic growth to $S_K$ and long-time dissipative behavior to the logarithmic relation, applicable to both chaotic and integrable systems. By analyzing various asymptotic forms of $b_n$ (linear, power-law, and logarithmic corrections), the work shows when the relation holds and how the prefactor $\tilde{\eta}$ depends on dynamical class, often satisfying $0<\tilde{\eta}\le 1$. The findings suggest that the $S_K$–$C_K$ relation encodes irreversibility in operator growth and could serve as a diagnostic of ergodicity and thermalization in quantum many-body dynamics.

Abstract

We study Krylov complexity $C_K$ and operator entropy $S_K$ in operator growth. We find that for a variety of systems, including chaotic ones and integrable theories, the two quantities always enjoy a logarithmic relation $S_K\sim \log{C_K}$ at long times, where dissipative behavior emerges in unitary evolution. Otherwise, the relation does not hold any longer. Universality of the relation is deeply connected to irreversibility of operator growth.

Figures (4)

• Figure 1: The functional relation $S_K\sim \mathrm{ln} C_K$ is shown for the model spectral density (\ref{['modelspectral']}), where $\nu=0\,,1\,,2$ from left to right. At long times, the linear relation Eq.(1) holds, with (shown in dashed lines) $S_K=0.976348\,\mathrm{ln}{C_K}+1.06661$ ($\nu=0$), $S_K=0.978129\,\mathrm{ln}{C_K}+0.934119$ ($\nu=1$) and $S_K=1.01102\,\mathrm{ln}{C_K}+0.63022$ ($\nu=2$). Within our numerical accuracy $\tilde{\eta}\simeq 1$ for all these cases. To have a nice presentation, we have moved the results along the horizontal axis properly.
• Figure 2: The functional relation $S_K(C_K)$ for nearly chaotic models. $b_n={{\frac{ \alpha n}{ \mathrm{ln}{(n+1)}} } }$ for the left panel and $b_n={{\frac{ \alpha n}{ \mathrm{ln}^2{(n+1)}} } }$ for the right panel. We set $\alpha=1$ in numerical calculations. At long times, the functional relation is very well fitted by $S_K=0.994787\,\mathrm{ln}{C_K}-0.958367\,\mathrm{ln}{\mathrm{ln}{C_K}}+1.04432$ (left) and $S_K=0.634549\,\mathrm{ln}{C_K}+0.593231\,\mathrm{ln}{\mathrm{ln}{C_K}}+0.59138$ (right), matching the continuum limit.
• Figure 3: Left panel: $b_n=\alpha\,\mathrm{ln}{(n+1)}$. Right panel: $b_n=\alpha\,\mathrm{ln}{n}+\gamma$. We set $\alpha=\gamma=1$. In both cases, the logarithmic relation (\ref{['skchaotic']}) already appears at the time scale where $C_K(t_c)\sim O(1)$ (see the orange lines) but the coefficient $\tilde{\eta}$ is changed at later times (see the red lines). Around $t\sim t_c$, we have $S_K=0.613581\,\mathrm{ln}{C_K}+1.35088$ (left) and $S_K=0.619693\,\mathrm{ln}{C_K}+1.36223$ (right) whereas at long times $S_K=0.166437\,\mathrm{ln}{C_K}+2.59585$ (left) and $S_K=0.170329\,\mathrm{ln}{C_K}+2.40017$ (right). Within our numerical accuracy, both cases have the same coefficient $\tilde{\eta}$ at the same time regimes.
• Figure 4: Trajectories on $C_K$-$S_K$ plane for finite chains. Left panel: $K=1$. Middle panel: $K=3$ and $\omega_2=2\omega_1$. In both cases, the particle moves periodically between $O$ and $F$. In the first half period of $C_K$, $t\in [0\,,T_C/2]$, it moves to the right, from $O$ to $F$ whereas in the next half period $t\in [T_c/2\,,T_C]$, it moves in the opposite direction, similar to a harmonic oscillator. Right panel: $K=3$ and $\omega_2=\sqrt{3}\omega_1$. The motion of the particle is not periodic and the trajectory becomes more and more complex as time increases. Here we have chosen $t\in [0\,,20]$.