Krylov Localization and suppression of complexity

TL;DR

The paperScreen investigates Krylov complexity as a diagnostic of late-time quantum dynamics, focusing on interacting integrable models. By mapping operator growth to a one-dimensional Krylov chain with off-diagonal disorder in the Lanczos coefficients, the authors show a localization mechanism that suppresses Krylov complexity saturation compared to chaotic systems. Using the XXZ spin chain as a testbed, they demonstrate left-biased diffusion on the Krylov chain and sublinear long-time C_K values, consistent with an Anderson-like localization picture. They further bolster this with a phenomenological model of disordered Lanczos sequences that reproduces the observed chaotic vs integrable behavior, highlighting the role of spectral statistics in shaping complexity growth and linking these findings to potential holographic interpretations.

Abstract

Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: This model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.

Figures (13)

• Figure 1: The Lanczos sequences for the various XXZ cases studied in this paper. For the details of the XXZ model see Section \ref{['Section_XXZ']}. For details on the one-site operator used and further numerical results, see Section \ref{['Sect_Numerical_Results']}.
• Figure 2: Using Lanczos-sequence data for various XXZ and complex SYK$_4$ systems of different Krylov dimensions, the standard deviation of the logarithm of ratios of consecutive Lanczos coefficients is represented by $\sigma$ on the $y$-axis, while the $x$-axis represents the Krylov dimension. The smoother the Lanczos-sequence is, the closer $\sigma$ is to zero. The lines connect the average values of each set of points for a given system size and are shown to help the reader. For each system size, each point represents a single realization of SYK and a single $J_{zz}$ value of XXZ. In particular we present the following data. SYK: 3 realizations for $L=8$ fermions, 3 realizations for $L=9$ and 5 realizations for $L=10$ (their Lanczos sequences can be found in Rabinovici:2020ryf); XXZ: 3 different $J_{zz}$ coefficients for $N=9$ spins, 5 for $N=10$, 5 for $N=11$ and 3 for $N=12$. The specific values chosen for $J_{zz}$ for each system size can be found in Figure \ref{['fig:Lanczos_XXZ']}.
• Figure 3: Nearest-neighbor level spacing distributions for XXZ with a specific $J_{zz}$ coupling with $N=14$ spins in the sector $M=7$, $P=+1$, both with open and periodic boundary conditions. For comparison, we include the corresponding distributions for complex SYK$_4$ with $L^{(SYK)}=13$ sites and occupation number $N^{(SYK)}=6$, whose Hilbert space dimension is equal to that of the positive parity sector of the above-mentioned XXZ instance. All distributions are normalized so that the area under each curve is equal to one, and all energy differences have been normalized by the mean level spacing $\Delta$ in each system.
• Figure 4: Numerical results for the transition probabilities $Q_{0n}$ defined in (\ref{['Transition_Probability']}) and $\overline{C_K}$ of XXZ systems with 9, 10, 11 and 12 spins and comparison with complex SYK$_4$ systems of 8, 9 and 10 fermions. The vertical lines represent $\overline{C_K}$ in units of $K$ for each system.
• Figure 5: The spectral decomposition of the initial operator (top) and K-complexities of the individual Liouvillian eigenstates (bottom). Data for 4 random realization of cSYK$_4$ with 10 fermions are superimposed in the leftmost column. Data for 3 choices of $J_{zz}$ coupling for XXZ with $N=12,\, M=4,\, P=+1$ are superimposed in the second column from the left. Data for 3 different choices of $J_{zz}$ couplings for XXZ with $N=11,\, M=5,\, P=+1$ are shown in the third column from the left; and data for 5 different choices of $J_{zz}$ couplings for XXZ with $N=10,\, M=4,\, P=+1$ are shown superimposed in the right column. The frequencies are normalized according to $\Delta$ which is the mean level spacing computed for for each system separately. The dashed red line represents the value of $K^{-1}$ for each system.