A Universal Operator Growth Hypothesis (1812.08657v5)

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 $\alpha$ in generic systems, with an extra logarithmic correction in 1d. The rate $\alpha$ --- an experimental observable --- governs the exponential growth of operator complexity in a sense we make precise. This exponential growth even prevails beyond semiclassical or large-$N$ limits. Moreover, $\alpha$ 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 $\lambda_L \leq 2 \alpha$, which complements and improves the known universal low-temperature bound $\lambda_L \leq 2 \pi T$. 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.

• The paper introduces a universal hypothesis that Lanczos coefficients grow linearly with sequence index, redefining our understanding of operator complexity.
• It leverages the Lanczos algorithm with analytical proofs and numerical simulations in spin-chain and SYK models to validate the linear growth pattern.
• The findings offer a practical bound on quantum chaos by linking the growth rate α to Lyapunov exponents, guiding future research in chaotic dynamics.

An Analysis of the Universal Operator Growth Hypothesis

The paper explores the dynamics of operators in many-body quantum systems under Hamiltonian evolution, proposing a Universal Operator Growth Hypothesis. It postulates that in generic, non-integrable systems, the Lanczos coefficients, derived from the continued fraction expansion of Green's functions, grow linearly with the sequence index, signifying an inherent universality in operator evolution. This notion stipulates a linear expansion rate denoted by α, signifying the rate at which the complexity of an operator grows exponentially.

Theoretical and Mathematical Framework

The authors establish their hypothesis within the context of the Lanczos algorithm, which converts linear-response calculations into a combinatorial problem involving a sequence of Lanczos coefficients. The core of the hypothesis states that these coefficients grow linearly as $b_n = \alpha n + \gamma + o(1)$ across most systems, with the allowance for a logarithmic modification in one-dimensional systems. This asymptotic growth allows α to serve as an upper bound for various complexity measures of operators, including out-of-time-order correlators (OTOCs), thereby offering a sharp bound on Lyapunov exponents $\lambda_L \leq 2\alpha$.

Numerical and Analytical Evidence

Empirical evidence supporting this hypothesis is drawn from several spin-chain models and the Sachdev-Ye-Kitaev (SYK) model—a well-studied model of chaotic systems with solvable large-N limits. Numerical simulations across various non-integrable systems reveal that the Lanczos coefficients adhere to a linear growth pattern, evidenced even as integrability-breaking terms are introduced into initially integrable models. Dramatically, in the SYK model, the paper corroborates the hypothesis analytically, providing exact calculations of the Lanczos coefficients in terms of the model's parameters. These theoretical predictions are supported by experimental observations, where high-frequency power spectra from historical quantum spin systems, such as CaF2, echoed the predicted exponential decay patterns consistent with the linear growth of the Lanczos coefficients.

Implications of the Growth Hypothesis

The hypothesis has significant implications for understanding quantum complexity and chaos. It portrays a foundational structure in which the growth rate α not only encapsulates the exponential increase of operator complexity but also provides a framework to understand universality in dynamical systems beyond the classical regime. The predictability conferred by the bound $\lambda_L \leq 2\alpha$ suggests that α encapsulates core information about chaotic growth in quantum systems, potentially bridging the gap between quantum and classical descriptions of chaos and complexity by providing a unified parameter to paper systems at varying degrees of quantum interaction.

Classical and Finite Temperature Extensions

The discussion extends to classical systems where the classical analog of operator complexity is explored. Here, a bound $\lambda_L \leq 4\alpha$ is conjectured in systems with extensive Lyapunov behavior. Moreover, the hypothesis is tested at finite temperatures in the SYK context, complementarily aligning with the universal bound on chaos offered by Maldacena, Shenker, and Stanford, thus establishing it as a potential general principle governing operator growth in quantum systems.

Conclusion and Future Directions

This analysis within the paper positions the Universal Operator Growth Hypothesis as a significant conjecture in theoretical physics. It challenges the existing paradigms about the nature of dynamics in quantum and chaotic systems and sets an agenda for future research. The direction suggested involves further exploration into non-integrable models to verify the universality of linear growth, and thorough investigations into how this theory can integrate with random matrix theory or other statistical mechanics frameworks. It also initiates vital discourse on the potential existence and identification of other q-complexities, urging the search for comprehensive theoretical principles applicable across both quantum and classical domains.