Krylov Localization and suppression of complexity

TL;DR

This paper explores how Krylov complexity behaves in integrable quantum systems, revealing a localization phenomenon that suppresses complexity growth compared to chaotic systems, with implications for understanding black hole dynamics.

Contribution

It introduces a novel localization mechanism on the Krylov chain in integrable models, linking complexity suppression to Anderson localization with off-diagonal disorder.

Findings

Krylov complexity saturation is suppressed in integrable models.

Localization on the Krylov chain is enhanced in integrable systems.

A phenomenological model reproduces the complexity behavior in different regimes.

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.