Krylov complexity in saddle-dominated scrambling

TL;DR

This work studies operator growth in the integrable Lipkin-Meshkov-Glick model to understand saddle-dominated scrambling. It shows that the Lanczos coefficients grow linearly, driving exponential Krylov complexity at early times, with the growth governed by a classical unstable saddle rather than chaos. A microcanonical extension reveals that the saddle dominates the exponential growth even away from its neighborhood, though the rate can differ elsewhere. The findings challenge the notion that exponential Krylov growth is a signature of chaos and motivate refining the universal operator growth hypothesis to include saddle-induced effects.

Abstract

In semi-classical systems, the exponential growth of the out-of-timeorder correlator (OTOC) is believed to be the hallmark of quantum chaos. However,on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.

Figures (9)

• Figure 1: A rough sketch of the behavior of Lanczos coefficients $b_{n}$ with $n$ for a "chaotic" system. The linear increase (blue) of Lanczos coefficients corresponds to the exponential growth of K-complexity $K(t)$ up to the time-scale $t_{*} = \ln \mathcal{S}$. After that, $b_n$ saturates to some constant (red), which corresponds to the linear increase of K-complexity. Finally, the decrease of $b_n$ (purple) is the Lanczos descent which marks the saturation of K-complexity.
• Figure 2: Behavior of OTOC for quantum LMG model (log plot). The OTOC is calculated for spin values $S = 25, 50$ and $75$. The early time behavior of OTOCs are fitted to an exponential function $\sim e^{\lambda_{\text{OTOC}}t}$. We obtain $\lambda_{\text{OTOC}} = 1.73446 > \lambda_{\text{saddle}} = \sqrt{3}$, which satisfies the classical bound \ref{['cbound']} on $\lambda_{\text{OTOC}}$.
• Figure 3: (a) Behavior of Lanczos coefficients $b_{n}$ with $n$. At small $n$, the growth of $b_{n}$ is linear and indistinguishable for all spins. After some $n$ the higher spin saturates to a larger value of $b_{n}$ before eventually falling to zero. This is a consequence of the finite size of the system (b) Behavior of K-complexity (log-plot) with time. At early time growth is exponential, as expected from a chaotic system. After a time scale of the order of system size, the growth reduces to a power-law nature, before eventually saturating (at large times). While it is not evident from the figure, it can be seen that the K-complexities for higher spins saturate at higher values.
• Figure 4: (a) Behavior of the Lanczos coefficients for LMG model with spin $S = 75$ and for $J = 5, 10, 15, 20, 25$. The straight lines have the equation $b_{n} = \alpha n + \beta$. The coefficient $\alpha$ follows the inequality $\alpha \geq \sqrt{2 J -1}/2$. (b) Growth of $b_n$ with $n$ in log-plot for $S=25$, and $J = 2$ till the edge of Krylov space.
• Figure 5: (a) Behavior of $2\alpha$ with respect to the energy $E$ plotted for $J = 2$ (blue). The $2\alpha (E = E_{\text{saddle}}) = \sqrt{2 J -1} = \sqrt{3}$ is given in orange. As is evident, the inequality $\sup_{E} 2\alpha(E) \geq \sqrt{2 J - 1}$ is saturated in this case. (b) Behavior of $2\alpha$ with respect to the energy $E$ plotted for $J = 1$ (blue). The $2\alpha (E = E_{\text{saddle}}) = \sqrt{2 J -1} = 1$ is given in orange. The inequality holds, and is not saturated. (c) and (d) Behavior of $2\alpha$ with respect to the energy $E$ plotted for $J = 3$ and $10$ respectively (blue). The $2\alpha (E = E_{\text{saddle}}) = \sqrt{2 J -1} = \sqrt{5}$ and $\sqrt{19}$ respectively, and are given in orange. The inequality holds, and is not saturated. In fact, we can see that the difference between the two values increases with $J$. Note that this happens for $J > 2$. For $1/2 <J < 2$, the difference decreases as $J$ increases until it goes to $0$ at $J = 2$.