A Universal Operator Growth Hypothesis

Abstract

We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate α in generic systems, with an extra logarithmic correction in 1d. The rate α -an experimental observable -governs the exponential growth of operator complexity in a sense we make precise. This exponential growth prevails beyond semiclassical or large-N limits. Moreover, α upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, we obtain a sharp bound on Lyapunov exponents λL ≤ 2α, which complements and improves the known universal low-temperature bound λL ≤ 2πT . We illustrate our results in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants. CONTENTS

Conformal field theory and the web of quantum chaos diagnostics

“…(10)(11)(12) The rate of growth of Lanczos coefficients, a notion of operator complexity was shown to bound the Lyapunov exponent of OTOCs in Ref. [16]. Both the bounds on this rate and the Lyapunov exponent were shown to follow from the eigenstate thermalization hypothesis in Ref.…”

“…[8]. (15)(16) The entanglement content of a Heisenberg time-evolved local operator may be shown to be related to relative entropy of excited states and the OTOC of the local operator with twist fields [18]. separated eigenvalues.…”

On the evolution of operator complexity beyond scrambling

“…The authors of ref. [1] have proposed a variant of these notions which uses an operator basis adapted to the time evolution of operators, rather than an a priori basis of the operator algebra. Starting from some initial operator O 0 , one can envision the Heisenberg flow on the space of operators, O t = e itH O 0 e −itH , whose Taylor expansion with respect to the time variable is generated by the set of nested commutators of O 0 with the Hamiltonian.…”

“…Using these nested commutators as linear generators of the operator algebra one can describe the Heisenberg flow as gradually accesing a growing subspace of the operator space. K-complexity is defined in [1] as an effective dimension of this growing subspace.…”

“…Using the SYK model as a benchmark model for a fast scrambler, it has been shown in [1] that K-complexity is similar to other definitions of operator size, when working in the thermodynamic limit. In this paper, we move away from the thermodynamic limit and study the regime of very long times, much larger than the scrambling time, when operator size ceases to be a useful characterization of complexity.…”