Abstract
We study operator complexity on various time scales with emphasis on those much larger than the scrambling period. We use, for systems with a large but finite number of degrees of freedom, the notion of K-complexity employed in [1] for infinite systems. We present evidence that K-complexity of ETH operators has indeed the character associated with the bulk time evolution of extremal volumes and actions. Namely, after a period of exponential growth during the scrambling period the K-complexity increases only linearly with time for exponentially long times in terms of the entropy, and it eventually saturates at a constant value also exponential in terms of the entropy. This constant value depends on the Hamiltonian and the operator but not on any extrinsic tolerance parameter. Thus K-complexity deserves to be an entry in the AdS/CFT dictionary. Invoking a concept of K-entropy and some numerical examples we also discuss the extent to which the long period of linear complexity growth entails an efficient randomization of operators.
Highlights
JHEP10(2019)264 where c is a numerical constant of O(1), and we neglect various polynomial corrections to this expression [4]
We study operator complexity on various time scales with emphasis on those much larger than the scrambling period
We present evidence that K-complexity of Eigenstate Thermalization Hypothesis (ETH) operators has the character associated with the bulk time evolution of extremal volumes and actions
Summary
We begin with a review of K-complexity and a description of the notational conventions to be used in this paper. Since ‘operator size’ is an intuitive measure of its complexity, and operator size is roughly related to the ordering in the Krylov basis, it is natural to define the notion of K-complexity as the average value of n in the Krylov basis expansion (2.7), i.e. where unit normalization of the φn amplitudes is assumed. A given pattern of growth of Lanczos coefficients as a function of n translates into a characteristic growth of complexity It is shown in [1] that a system with an asymptotic large-n law bn ≈ α n ,. It is natural to propose (2.11) as a criterion for local quantum chaos, since explicit evaluation of Lanczos coefficients in various integrable systems yield softer asymptotic laws of the form bn ∼ α nδ , 0 < δ < 1 In these cases, K-complexity has a milder, powerlike growth: CK(t) ∼ (αt) 1−δ. An asymptotic power-law behavior of the moments is associated with a flat distribution of Lanczos coefficients
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