Probing the localization effects in Krylov basis

TL;DR

This work investigates how localization phenomena in the quantum kicked rotor manifest in the Krylov basis through Krylov complexity ($\mathcal{C}(t)$) and Arnoldi coefficients. Using Floquet dynamics and Arnoldi iteration, the authors build a K-basis for two QKR variants (CQKR and SQKR) and analyze four localization regimes: quantum anti-resonance (AR), classical-induced localization (CIL), dynamical localization (DL), and power-law localization (PL). They show that $\mathcal{C}(t)$ and $h_{n,n-1}$ exhibit regime-specific growth and saturation patterns, enabling differentiation between classical and quantum localization, while the variance of Arnoldi coefficients more reliably signals the integrability-to-chaos transition than $\mathcal{C}(t)$. The results highlight the Krylov basis as a powerful lens for localization dynamics in Floquet systems and point to extensions toward interacting and many-body localization scenarios.

Abstract

Krylov complexity (K-complexity) is a measure of quantum state complexity that minimizes wavefunction spreading across all the possible bases. It serves as a key indicator of operator growth and quantum chaos. In this work, K-complexity and Arnoldi coefficients are applied to probe a variety of localization phenomena in the quantum kicked rotor system. We analyze four distinct localization scenarios -- ranging from compact localization effect arising from quantum anti-resonance to a weaker form of power-law localization -- each one exhibiting distinct K-complexity signatures and Arnoldi coefficient variations. In general, K-complexity not only indicates the degree of localization, but surprisingly also of the nature of localization. In particular, the long-time behaviour of K-complexity and the wavefunction evolution on Krylov chain can distinguish various types of observed localization in QKR. In particular, the time-averaged K-complexity and scaling of the variance of Arnoldi coefficients with effective Planck's constant can distinguish the localization effects induced by the classical regular phase structures and the dynamical localization arising from quantum interferences. Further, the Arnoldi coefficient is shown to capture the transition from integrability to chaos as well. This work shows how localization dynamics manifests in the Krylov basis.

Figures (10)

• Figure 1: Top row: Phase space plots of the standard kicked rotor and singular kicked rotor models. The x-axis represents the angular position $x$, while the y-axis shows the conjugate momentum $p$. (a) KR with $K = 0.5$, showing regular motion around primary islands, (b) KR with $K = 8$, exhibiting chaotic behaviour with large diffusion in phase space, (c) Singular KR with $K = 1$, $\alpha = 0.5$, demonstrating complete chaos in phase space. Bottom rows ((d)-(g)): Probability density $|\phi(m)|^2$ of the averaged eigenstates in momentum space for different localization regimes in the Log-scale. $m$ denotes the momentum index. (d) Quantum anti-resonance (AR) for $K = 0.5$, $\hbar_s = 2\pi$, (e) Classical-induced localization (CIL) for $K = 0.5$, $\hbar_s = 1$, (f) Dynamical localization (DL) for $K = 8$, $\hbar_s = 1$ (dotted line: exponential fit), (g) Power-law localization (PL) in the singular kicked rotor (SQKR) model with $K = 1$, $\alpha = 0.5$ (dotted line: power-law fit). (h) Delocalized eigenvector, representing a random complex eigenstate with uniform probability distribution in momentum space.
• Figure 2: Probability density $|\phi_{n}(t)|^{2}$ and inverse participation ratio (IPR) in the K-basis as a function of the number of kicks ($t$) for different parameter sets. (a) AR case, where red dots highlight strong localization at the first and second K-basis vectors. (b) CIL case, showing relatively weaker localization in the K-basis compared to the AR case. (c) DL case, with the black dotted line marking the break time, beyond which localization sets in. (d) Singular kicked rotor case with $K = 1$, displaying power-law localized states and corresponding IPR, indicating weak localization effects.
• Figure 3: Inverse participation ratio (IPR) in the K-basis as a function of $K$ for different types of localization in CQKR: (a) AR, (b) CIL, and (c) DL. Panel (d) shows the IPR as a function of $\alpha$ for power-law localization (PL) in the SQKR. Higher IPR values indicate stronger localization of the wave function in the K-basis. The red dotted lines in (c) and (d) serve as a guide to highlight the linear trend of the IPR with the respective parameters.
• Figure 4: K-complexity ($\mathcal{C}(t)$) and Arnoldi coefficients ($h_{n,n-1}(t)$) across the four regimes of localization: AR, CIL, DL, and PL. The left column shows the time evolution of K-complexity while the right column illustrates the behaviour of Arnoldi coefficients corresponding to each localization regime.
• Figure 5: Comparison of average K-complexity ($\overline{\mathcal{C}}$) and the variance of Arnoldi coefficients $\sigma^{2}(h_{n,n-1})(t)$ as indicators of the integrability-to-chaos transition. (a) shows the growth of average K-complexity with respect to $K$ and (b) displays the variance of Arnoldi coefficients with respect to $K$.