A Universal Operator Growth Hypothesis

TL;DR

The paper proposes a universal operator growth hypothesis: in generic non-integrable many-body systems, Lanczos coefficients grow linearly with index, with a 1D logarithmic correction, and the growth rate $\alpha$ governs exponential operator growth and bounds a broad class of operator complexities, including OTOCs. It connects high-frequency spectral tails to the Lanczos data, derives bounds on the Lyapunov exponent $\lambda_L\le 2\alpha$, and provides analytic and numerical evidence from SYK, spin chains, and classical chaos. The work further develops a practical method to compute diffusion constants via a meromorphic continuation of the Green's function using the continued fraction expansion, and extends the framework to finite temperatures with a conjectured finite-$T$ chaos bound. Overall, the results offer a coherent, testable picture linking operator growth, chaos, hydrodynamics, and diffusion in quantum many-body dynamics, with potential implications for both theory and experiment.

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 even 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 $λ_L \leq 2 α$, which complements and improves the known universal low-temperature bound $λ_L \leq 2 πT$. 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.

Figures (10)

• Figure 1: Artist's impression of the space of operators and its relation to the 1d chain defined by the Lanczos algorithm starting from a simple operator $\mathcal{O}$. The region of complex operators corresponds to that of large $n$ on the 1d chain. Under our hypothesis, the hopping amplitudes $b_n$ on the chain grow linearly asymptotically in generic thermalizing systems (with a log-correction in one dimension, see Section \ref{['sec:1dspecial']}). This implies an exponential spreading $\mathinner{\left(n\right)}_t \sim e^{2\alpha t}$ of the wavefunction $\varphi_n$ on the 1d chain, which reflects the exponential growth of operator complexity under Heisenberg evolution, in a sense we make precise in Section \ref{['sec:complexity']}. The form of the wavefunction $\varphi_n$ is only a sketch; see Fig. \ref{['fig:exponential_spreading_wavefunction']} for a realistic picture.
• Figure 2: Lanczos coefficients in a variety of models demonstrating common asymptotic behaviors. "Ising" is $H = \sum_i X_i X_{i+1} + Z_i$ with $\mathcal{O} = \sum_j e^{iq_j} Z_j$ ($q = 1/128$ here and below) and has $b_n \sim{} O(1)$. "X in XX" is $H = \sum_i X_i X_{i+1} + Y_i Y_{i+1}$ with $\mathcal{O} = \sum_j X_j$, which is a string rather than a bilinear in the Majorana fermion representation, so this is effectively an interacting integrable model that has $b_n \sim{} \sqrt{n}$. XXX is $H = \sum_i X_i X_{i+1} + Y_i Y_{i+1} + Z_i Z_{i+1}$ with $\mathcal{O} = \sum_j e^{iq_j} (X_j Y_{j+1} - Y_j X_{j+1})$ that appears to obey $b_n \sim{} \sqrt{n}$. Finally, SYK is \ref{['eq:Q_SYK_model']} where $q=4$ and $J= 1$ and $\mathcal{O} = \sqrt{2}\gamma_1$ with $b_n \sim{} n$. The Lanczos coefficients have been rescaled vertically for display purposes. Numerical details are given in Appendices \ref{['app:SYK']} and \ref{['app:numerics']}.
• Figure 3: Illustration of the spectral function and the analytical structure of $C(t), t \in \mathbb{C}$. When the Lanczos coefficients have linear growth rate $\alpha$, $\Phi(\omega)$ has exponential tails $\sim e^{-|\omega|/\omega_0}$ with $\omega_0 = 2 \alpha / \pi$; $C(t)$ is analytical in a strip of half-width $1/\omega_0$ and the singularities closest to the origin are at $t = \pm i / \omega_0$. See Appendix \ref{['app:recursion_2']} for further discussion.
• Figure 4: (a) Lanczos coefficients in a variety of strongly interacting spin-half chains: $H_1 =\sum_i X_iX_{i+1} + 0.709 Z_i + 0.9045 X_i$, $H_2 = H_1 + \sum_i 0.2 Y_i$, $H_3 = H_1 + \sum_i 0.2 Z_i Z_{i+1}$. The initial operator $\mathcal{O}$ is energy density wave with momentum $q=0.1$. (b) Cross-over to apparently linear growth as interactions are added to a free model. Here $H= \sum_{i} X_i X_{i+1} -1.05 Z_i + h_X X_i$, and $\mathcal{O} \propto \sum_i 1.05 X_i X_{i+1} + Z_i$. The $b_n$'s are bounded when $h_X = 0$ but appears to have asymptotically linear growth for any $h_X \neq 0$. Logarithmic corrections are not clearly visible in the numerical data. Numerical details are given in Appendix \ref{['app:numerics']}.
• Figure 5: The exact solution wavefunction \ref{['eq:universal_wavefunction_main_text']} in the semi-infinite chain at various times. The wavefunction is defined only at $n=0,1,2\dots$, but has been extrapolated to intermediate values for display.