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.
Highlights
Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems
In this paper we studied Lanczos coefficients and operator growth in Krylov space for local operators in various CFT models
A general argument presented in the introduction dictates that so far asymptotic behavior of bn as a function of n is sufficiently smooth, Lanczos coefficients exhibit universal operator growth hypothesis (10) and Krylov complexity grows exponentially (14)
Summary
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). In the context of field theory and large N models another well-studied signature of chaos is the behavior of the out of time ordered correlator (OTOC) [2] These approaches focus on different aspects of quantum dynamics and usually apply to different systems. From one side connection of Krylov complexity to OTOC is not that surprising given that the latter measures spatial operator growth [7] From another side, dynamics in Krylov space is fully determined in terms of thermal 2pt function, see below. To conclude the introductory part, we remark that studying Krylov complexity should be seen in a broader context of relating it to holographic complexity [10,11,12] and studies of thermal 2pt function in holographic settings with the goal of elucidating quantum gravity in the bulk [13,14,15,16,17,18,19,20]
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