[Paper Review] Constraints on hydrodynamics from many-body quantum chaos

Andrew Lucas|arXiv (Cornell University)|Oct 3, 2017

Quantum, superfluid, helium dynamics71 references16 citations

TL;DR

This paper establishes fundamental bounds on hydrodynamic transport in quantum many-body systems by linking hydrodynamics to many-body quantum chaos via the butterfly effect. It derives upper limits on sound speed and diffusion constants using the butterfly velocity, showing that hydrodynamics cannot propagate faster than quantum information scrambling, with key results $v_s \leq v_B$ and $D \leq v_B^2 \tau$ in systems with $N \sim 1$ degrees of freedom per site.

ABSTRACT

Is the hydrodynamics of an interacting many-body system fundamentally limited by basic principles of quantum mechanics? Starting with the conjecture that viscosity is at least as large as entropy density (as measured in fundamental units), there has been a long search for a precise answer to this question. In this work, we identify a simple relationship between hydrodynamics and many-body quantum chaos in a broad class of experimentally realizable systems. Consistency with the quantum butterfly effect leads to upper bounds on the speed of sound and diffusion constants of hydrodynamics. These bounds link two very different theories of quantum many-body dynamics, clarify the relationship between classical hydrodynamics and quantum information loss, and provide a simple way to constrain theories of thermalization and quantum chaos in experiments.

Motivation & Objective

• To determine whether hydrodynamics in quantum many-body systems is fundamentally constrained by quantum mechanics.
• To investigate the relationship between hydrodynamic transport (diffusion and sound propagation) and the spread of quantum information via the butterfly effect.
• To derive universal upper bounds on diffusion constants and sound speeds using the butterfly velocity as a fundamental scale.
• To resolve apparent paradoxes in large-$N$ theories where hydrodynamics appears to outpace scrambling by generalizing bounds to multiple scrambling velocities.
• To provide experimentally testable constraints on thermalization and quantum chaos in strongly correlated systems.

Proposed method

• Defines the butterfly velocity $v_B$ as the minimal speed at which quantum information spreads, based on out-of-time-ordered correlators (OTOCs) decaying as $\mathcal{G}(|x| - v_B t)$.
• Uses the conjecture that hydrodynamics must be consistent with the butterfly effect, implying that diffusion and sound cannot propagate faster than $v_B$.
• Applies holographic methods in $\mathrm{AdS}_{d+2}$-like geometries to compute quasinormal modes and extract diffusion constants $D$ and relaxation times $\tau_*$.
• Derives bounds via matching solutions to the equation of motion in the hydrodynamic and near-horizon regimes, enforcing infalling boundary conditions.
• Estimates $D \lesssim v_B^2 \tau_*$ by comparing the timescale $\tau_*$ for quantum information to spread to the scale of diffusion.
• Generalizes bounds to large-$N$ systems by identifying multiple scrambling velocities and showing that $v_s \leq v_B$ and $D \leq v_B^2 \tau$ still hold.

Experimental results

Research questions

• RQ1 Can hydrodynamic transport be fundamentally limited by the speed of quantum information scrambling?
• RQ2 What is the precise relationship between the butterfly velocity $v_B$ and the diffusion constant $D$ in quantum many-body systems?
• RQ3 Why do some holographic models exhibit hydrodynamic modes at frequencies $\sim -i T$ rather than $\sim -i v_B^2 / D$?
• RQ4 How do the bounds on $D$ and $v_s$ generalize in large-$N$ theories where multiple scrambling velocities exist?
• RQ5 Can the absence of a non-quasinormal mode at $\omega \sim -i v_B^2 / D$ be reconciled with hydrodynamic effective field theory?

Key findings

• The sound speed is bounded by the butterfly velocity: $v_s \leq v_B$, ensuring hydrodynamics cannot propagate faster than quantum information scrambling.
• The diffusion constant satisfies $D \leq v_B^2 \tau$, where $\tau$ is the timescale for quantum information to spread, derived from the condition that scrambling must precede diffusion.
• In charge-neutral Lifshitz theories with $z > d$, the smallest quasinormal mode occurs at $\omega \sim -i T$, implying $\tau_* \ll \tau$, and $D \lesssim v_B^2 \tau_*$ is tighter than $D \lesssim c^2 \tau_*$.
• The bound $D \leq v_B^2 \tau$ is parametrically stronger than previous bounds based on the Lieb-Robinson velocity $v_{\mathrm{LR}}$, especially when $v_B \ll v_{\mathrm{LR}}$.
• In holographic models, the absence of a quasinormal mode at $\omega \sim -i v_B^2 / D$ indicates that the standard hydrodynamic form $\omega \sim -i D(k^2 - \omega^2 / v_B^2)$ does not capture the true low-energy dynamics.
• The bound $D \lesssim c^2 \tau_*$, where $c \sim v_{\mathrm{LR}}$, is found to be weak compared to $D \lesssim v_B^2 \tau_*$, suggesting it may be a model-dependent artifact rather than a universal constraint.