A Universal Operator Growth Hypothesis

By D. Parker, Xiangyu Cao, Thomas Scaffidi, E. Altman. Physical Review X, 2018

513 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/687b5300168353afd627fdec2a59e30a2851fe22

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Citations in the corpus

Krylov complexity in saddle-dominated scrambling (2022, methodology): “This in turn suggests that C(t) with t ∈ C has pole at purely imaginary axis at location t = ±iπ/2α (nearest to the origin) [36].”
Krylov complexity in conformal field theory (2021, methodology): “Dynamics of quantum operators in Krylov space has been recently proposed as a potential bridge connecting dynamics of OTOC with the conventional signatures of many-body chaos [3].”
Operator complexity: a journey to the edge of Krylov space (2020, background): “A notion of complexity known as Krylov complexity, or K-complexity, was introduced in [9] as a useful diagnostic of quantum chaos in the thermodynamic limit.”
On the evolution of operator complexity beyond scrambling (2019, methodology): “The main reference for this section is [1].”
Scrambling Dynamics and Out-of-Time-Ordered Correlators in Quantum Many-Body Systems (2022, unknown): “In this section, we discuss the microscopic physics of OTOCs in terms of the growth of Heisenberg operators [25, 26].”
Geometry of Krylov complexity (2021, methodology): “Finally, the idea of Krylov complexity, that will be our main topic, was put forward in [23].”
Krylov complexity from integrability to chaos (2022, methodology): “O(E) and S(E) are the microcanonical one-point function and entropy, respectively, and the function fO(E,ω) gives the Fourier transform of the connected two-point function, sometimes denoted spectral function [3].”
Quantum chaos and the complexity of spread of states (2022, unknown): “Starting with [47], recent literature [48–63] has mainly focused on the Wightman inner product”