Krylov complexity in large-$q$ and double-scaled SYK model

TL;DR

<3-5 sentence high-level summary> This paper investigates operator growth and quantum chaos in SYK-type models through the lens of Krylov complexity. It develops and analyzes Lanczos coefficients and Krylov cumulants in the large-$q$ expansion (two-stage limit) up to $O(1/q^2)$, elucidating how subleading corrections reshape complexity growth and its fluctuations. It then studies the double-scaled, infinite-temperature limit (DSSYK$_{∞}$) where $q orac{N^{1/2}}$, showing hyperfast scrambling with a scrambling time that vanishes as $q o\infty$ and divergent Lanczos coefficients, with implications for nonlocal holography and de Sitter space. The results highlight the sensitivity of Krylov-based probes to locality assumptions and offer a framework for exploring scrambling across distinct scaling limits in SYK-like systems.

Abstract

Considering the large-$q$ expansion of the Sachdev-Ye-Kitaev (SYK) model in the two-stage limit, we compute the Lanczos coefficients, Krylov complexity, and the higher Krylov cumulants in subleading order, along with the $t/q$ effects. The Krylov complexity naturally describes the "size" of the distribution, while the higher cumulants encode richer information. We further consider the double-scaled limit of SYK$_q$ at infinite temperature, where $q \sim \sqrt{N}$. In such a limit, we find that the scrambling time shrinks to zero, and the Lanczos coefficients diverge. The growth of Krylov complexity appears to be "hyperfast", which is previously conjectured to be associated with scrambling in de Sitter space.

Figures (6)

• Figure 1: (a) Comparison of the auto-correlation function $\varphi_0 (t)$ up to the $O(1/q)$ and $O(1/q^2)$ for $q=500$ (red). Inset shows the comparison for $q=100$. The dashed line gives the analytic result \ref{['diss']}. (b) The Lanczos coefficients $b_n$ in the large $q$ limit of the SYK model, keeping the $O(1/q)$ correction to the $b_n$s. We set $\mathcal{J} = 1$.
• Figure 2: (a) The different time-scale up to where the perturbation theory \ref{['Green1']} is valid. The blue, red and the green dots indicate $|g(t)| \sim q$, $|h(t)| \sim q^2$ and $|h(t)| \sim q \,|g(t)|$ respectively. To compute the $t_c$, we choose the last one, namely $|h(t_c)| \sim q \,|g(t_c)|$. (b) The difference $d_{n} \equiv |\varphi^{(1/q^2)}_{n}(t) - \varphi^{(1/q)}_{n}(t)|$ is plotted, between the Krylov wavefunctions at the $1/q$ and $1/q^2$ orders for $q = 100$. The plots correspond to $d_{1}, d_{2}, d_{3}$ and $d_{4}$. As is seen, $d_{n}$ increases significantly (odd wavefunctions increase negatively while the even wavefunctions increase positively) well before the cutoff time.
• Figure 3: (a) The behavior of truncated K-complexity up to $\mathcal{O}(1/q^{2})$ for $q = 500$. The dashed line is the analytic result \ref{['cc']}. The inset shows the early time behavior. The dotted lines represent the time $t_{c}$ up to which our results are reliable. (b) Various $\varphi_n(t)$ is shown for $q=100$. These are obtained by taking the auto-correlation function \ref{['Green1']} and the Lanczos coefficients \ref{['bnsubleading']}. These wave functions are evaluated without any truncation in $q$. The dashed line indicates the analytic result \ref{['diss']}. We observe that the matching of odd wavefunctions is better than the even wavefunctions.
• Figure 4: The behavior of K-complexity is shown for (a) $q=100$ and (b) $q = 500$ respectively, with the Lanczos coefficients \ref{['bnsubleading']}. The dashed line demonstrates the analytic result \ref{['cc']}.
• Figure 5: The behavior of K-entropy is shown for (a) $q=100$ and (b) $q = 500$ respectively, with the Lanczos coefficients \ref{['bnsubleading']}. The dashed line demonstrates the analytic result \ref{['cent']}.