Krylov complexity in conformal field theory

TL;DR

This paper investigates Krylov complexity as a chaos probe in various conformal field theories, revealing that it can grow exponentially even in non-chaotic models, challenging previous assumptions about chaos indicators.

Contribution

It demonstrates that Krylov complexity can grow exponentially in free and rational CFTs, contradicting the expectation that exponential growth indicates chaos.

Findings

Krylov complexity bounds the chaos measure OTOC.

Exponential growth of Krylov complexity observed in non-chaotic CFTs.

Challenges the use of exponential Krylov complexity as a chaos indicator.

Abstract

Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems. It is a measure of operator growth in Krylov space, which conjecturally bounds the operator growth measured by the out of time ordered correlator (OTOC). We study Krylov complexity in conformal field theories by considering arbitrary 2d CFTs, free field, and holographic models. We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford. In all considered examples including free and rational CFTs Krylov complexity grows exponentially, in stark violation of the expectation that exponential growth signifies chaos.