On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

TL;DR

The bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces is resort to, which renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and chaotic systems by computing Krylov complexity at late times.

Abstract

Continuing the previous initiatives arXiv: 2207.05347 and arXiv: 2212.06180, we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and chaotic systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

Figures (9)

• Figure 1: (a) Growth of Lanczos coefficients $b_n$ for the probe operator $\sigma_3^z$ in integrable ($g=1$, $h=0$, red) and chaotic ($g=-1.05$, $h=0.5$, blue) limits as mentioned below Eq. \ref{['tfim']} by implementing the bi-Lanczos algorithm. The system size is $N = 6$. Since this is a closed system, the Lanczos and bi-Lanczos algorithm yields the same results. The behavior of the first few coefficients is shown in the inset image. In the Lanczos descent, there are more fluctuations for the integrable case compared to the non-integrable one. (b) The behavior of K-complexity for integrable (red) and chaotic (blue) limits. Observe that the saturation value in the chaotic limit is higher ($\sim \mathcal{K}/2$, dashed black line) than the saturation value in the integrable limit.
• Figure 2: Growth of Lanczos coefficients for (a) integrable ($g=1$, $h=0$) and (b) chaotic ($g=-1.05$, $h=0.5$) systems for $\alpha=0.01, \, \gamma=0.01$. The system size is $N = 6$. The yellow and cyan lines in the respective plots show the averaged values, done only to better understand the overall Lanczos descent. One can easily see that in the case of open systems, even the chaotic $b_n$ have fluctuations for large $n$ resulting in the same saturation value of K-complexity as the integrable case. The information of integrability, therefore, washes out at higher times, or higher $n$. (c) shows the comparison between $|a_n|$ plots for integrable and non-integrable cases with $\alpha=0.01$, $\gamma=0.01$.
• Figure 3: Behavior of $\eta$ with $\alpha$ and $\gamma$, according to Eq.\ref{['eta']}. In both cases, the plots show linear growth. We use values from Table \ref{['table1']} (a) and (b).
• Figure 4:
• Figure 5: Comparison of K-complexities between the integrable ($g=1$, $h=0$, red) and the non-integrable ($g=-1.05$, $h=0.5$) for fixed $\alpha=0.05$ with (a) $\gamma=0$ and (b) $\gamma=0.01$ and (c) $\gamma=0.05$.