Krylov complexity from integrability to chaos

TL;DR

This work investigates Krylov (K-)complexity as a diagnostic of the transition from integrability to chaos in the XXZ spin chain using integrability-breaking deformations. By mapping operator spreading to a Krylov chain and incorporating an off-diagonal Anderson-hopping picture, the authors connect late-time K-complexity saturation to disorder in the Lanczos sequence and compare chaotic regimes to random-matrix theory (RMT) in the appropriate symmetry class. They analyze how the seed operator and ETH structure influence KC_sat, showing that chaotic systems approach RMT predictions while integrable dynamics yield significantly lower saturation values. The results highlight the role of operator structure, spectral statistics, and Thouless energy in governing Krylov-space spreading, with implications for understanding chaos and potential holographic connections.

Abstract

We apply a notion of quantum complexity, called "Krylov complexity", to study the evolution of systems from integrability to chaos. For this purpose we investigate the integrable XXZ spin chain, enriched with an integrability breaking deformation that allows one to interpolate between integrable and chaotic behavior. K-complexity can act as a probe of the integrable or chaotic nature of the underlying system via its late-time saturation value that is suppressed in the integrable phase and increases as the system is driven to the chaotic phase. We furthermore ascribe the (under-)saturation of the late-time bound to the amount of disorder present in the Lanczos sequence, by mapping the complexity evolution to an auxiliary off-diagonal Anderson hopping model. We compare the late-time saturation of K-complexity in the chaotic phase with that of random matrix ensembles and find that the chaotic system indeed approaches the RMT behavior in the appropriate symmetry class. We investigate the dependence of the results on the two key ingredients of K-complexity: the dynamics of the Hamiltonian and the character of the operator whose time dependence is followed.

Figures (12)

• Figure 1: This table summarizes the general behaviour expected from K-complexity, particularly for finite chaotic systems with $S$ degrees of freedom ($\Lambda$ is the bandwidth of the system). It is based on Parker:2018yvkBarbon:2019wsy and Rabinovici:2020ryf.
• Figure 2: Left: Probability distribution functions for the $\tilde{r}$ statistics of (\ref{['XXZ+Hd']}) (top) and (\ref{['XXZ+NNN']}) (bottom) with increasing value of $\epsilon_d$ and $J^{(2)}_{zz}$ respectively, computed for $N=19$ spins in the $M=7, P=+1$ sector with $J_{zz}=1.1$. The purple line represents the analytical result for $P(\tilde{r})$ in the case of Poissonian level-spacing statistics, while the yellow line represents the analytical result for GOE ensembles. Right: The value of $\langle\tilde{r}\rangle$ as a function of $\epsilon_d$ (top) and $J^{(2)}_{zz}$ (bottom). Horizontal lines represent analytical values for Poisson (purple) and GOE (yellow). The colored dots represent point for which we plotted the $P(\tilde{r})$ distribution function in the left panel, while the gray dots represent additional data points.
• Figure 3: Distribution of the log of consecutive ratios of Lanczos coefficients. Inset: Standard-deviation $\sigma$ of this distribution as a function of the corresponding integrability breaking term. The standard deviation generally decreases with the coefficient of the integrability breaking term. Comparing with the corresponding computations of the K-complexity saturation values in Figures \ref{['fig:N11M6Hd']} and \ref{['fig:N11M4J2zz']}, this is consistent with the phenomenology described in Rabinovici:2021qqt namely that the saturation values of K-complexity will increase with decreasing disorder in the Lanczos coefficients.
• Figure 4: Late-time transition probability results for local operator of the form (\ref{['Operator']}) with trace removed, for $H_{XXZ}$ with an $H_d$ integrability-breaking term. The vertical lines represent the late-time saturation value of KC as a fraction of the Krylov space dimension. Left: For $N=11$ spins in the sector $M=4, P=+1$ with $J_{zz}=1.1$ for the operator $\mathcal{O}=S_6^z$. The Krylov space dimension is $K=28731$. Right: For $N=11$ spins in the sector $M=6, P=+1$ with $J_{zz}=1.1$ for the operator $\mathcal{O}=S_5^z+S_7^z$. For this system the Krylov space dimension is $K=55461$. Inset: dependence of KC saturation value on the strength of the integrability-breaking term.
• Figure 5: Results for the saturation value of K-complexity computed for a local operator of the form (\ref{['Operator']}) with trace removed, for $H_{XXZ}+J^{(2)}_{zz}\,H_{NNN}$ integrability-breaking term in the sector $N=11, M=4, P=+1$. Left: With $J_{zz}=0.72$, for the operator $\mathcal{O}=S_5^z+S_7^z$. Right: With $J_{zz}=0.91$, for the operator $\mathcal{O}=S_6^z$. For both systems the Krylov space dimension is $K=28731$. Inset: dependence of KC saturation value on the strength of the integrability-breaking term.