A quantum hydrodynamical description for scrambling and many-body chaos

TL;DR

This work proposes a quantum hydrodynamic framework where a single hydrodynamic mode σ, tied to energy conservation, governs both scrambling and butterfly spreading in chaotic quantum many-body systems. A shift symmetry of the hydrodynamic action yields an exponentially growing mode that drives OTOC growth, while keeping energy-density and energy-flux correlators non-exponential and introducing the pole-skipping phenomenon in energy correlators. The theory unifies chaotic dynamics across holographic models and SYK-like chains, deriving ballistic spreading with butterfly velocity v_B from diffusion kernels and relating v_B to energy diffusion D_E and the Lyapunov exponent λ. It also provides practical diagnostics, such as extracting λ and v_B from pole-skipped energy-density correlators, offering a pathway to identify maximal chaos via hydrodynamic principles. The framework points to deep connections between chaos, hydrodynamics, and horizon physics, with potential extensions to momentum-conserving systems and broader chaotic regimes.

Abstract

Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in specific models. In this paper we propose a unified description of the exponential growth and ballistic butterfly spreading of OTOCs across different systems using a newly formulated "quantum hydrodynamics," which is valid at finite $\hbar$ and to all orders in derivatives. The scrambling of a generic few-body operator in a chaotic system is described as building up a "hydrodynamic cloud," and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic action. The shift symmetry also shields correlation functions of the energy density and flux, and time ordered correlation functions of generic operators from exponential growth, while leads to chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping of a pole at special values of complex frequency and momentum in two-point functions of energy density and flux. This pole-skipping phenomenon may be considered as a "smoking gun" for the hydrodynamic origin of the chaotic mode. We also discuss the possibility that such a hydrodynamic description could be a hallmark of maximally chaotic systems.

Figures (7)

• Figure 1: A cartoon of scrambling. The operator $V(t)$ expands in the space of degrees of freedom. $t_r$ is relaxation time, and $t_s$ is the scrambling time when $V(t)$ is essentially scrambled among all degrees of freedom.
• Figure 2: We propose that operator growth of $V(t)$ can be described in terms of a bare operator $\hat{V}(t)$ surrounded by an expanding hydrodynamic cloud.
• Figure 3: At leading order in large ${\cal N}$ correlation functions are controlled by exchange of hydrodynamic fields $\sigma(t)$. The only difference between time ordered (left) and out-of time-ordered (right) configurations is in the effective vertex describing the coupling to $\sigma(t)$.
• Figure 4: In holographic theories scrambling is described by the interaction of a particle with near-horizon degrees of freedom. The wavy line in the plot represents a stretched horizon while solid line the event horizon. As a particle falls into a black hole, from the perspective of an outside observer, the particle is "dissolved" into the thermal cloud of a stretched horizon. Such a process may be interpreted in the dual quantum field theory as building up a hydrodynamic cloud .
• Figure 8: The hydrodynamic origin of chaos predicts "pole-skipping" in the energy-energy correlation function: following the line of poles which starts at small $\omega, k$ as the energy diffusion pole along pure imaginary $k$ to a specific value of $k = i \lambda/v_B$ for which the pole would be at $\omega= i \lambda$, one finds the pole is not there! In the figure the solid line denotes the line of poles, and the open dot indicates that at that particular point the pole is skipped. The dashed line is the curve $\omega = - i D_E k^2$ which coincides with the pole line for small $\omega, k$.