Speed Limits and Scrambling in Krylov Space

TL;DR

This paper explores how Krylov complexity relates to quantum speed limits and level repulsion, revealing intricate dynamics in many-body systems and differences between integrable and chaotic regimes.

Contribution

It uncovers the nuanced relationship between operator speed limits, level repulsion, and scrambling in many-body systems, highlighting the impact of integrability breaking.

Findings

Enhanced level repulsion increases OQSLs in random matrices.

Breaking integrability initially speeds up operator evolution, then slows it down at higher levels.

Krylov basis operators effectively measure scrambling in chaotic systems, less so in integrable ones.

Abstract

We investigate the relationship between Krylov complexity and operator quantum speed limits (OQSLs) of the complexity operator and level repulsion in random/integrable matrices and many-body systems. An enhanced level-repulsion corresponds to increased OQSLs in random/integrable matrices. However, in many-body systems, the dynamics is more intricate due to the tensor product structure of the models. Initially, as the integrability-breaking parameter increases, the OQSL also increases, suggesting that breaking integrability allows for faster evolution of the complexity operator. At larger values of integrability-breaking, the OQSL decreases, suggesting a slowdown in the operator's evolution speed. Information-theoretic properties, such as scrambling, coherence and entanglement, of Krylov basis operators in many-body systems, are also investigated. The scrambling behaviour of these operators exhibits distinct patterns in integrable and chaotic cases. For systems exhibiting chaotic dynamics, the Krylov basis operators remain a reliable measure of these properties of the time-evolved operator at late times. However, in integrable systems, the Krylov operator's ability to capture the entanglement dynamics is less effective, especially during late times.