Bounds on transport from univalence and pole-skipping

TL;DR

This paper introduces new methods combining complex analysis and univalence to derive exact bounds on hydrodynamic transport coefficients, with applications to quantum chaos and holography.

Contribution

It develops rigorous, sharp bounds on hydrodynamic dispersion relations using univalence and pole-skipping, expanding the theoretical toolkit for transport phenomena.

Findings

Derived bounds on diffusivity and sound speed

Established connections between transport bounds and quantum chaos

Provided holographic examples validating the bounds

Abstract

Bounds on transport represent a way of understanding allowable regimes of quantum and classical dynamics. Numerous such bounds have been proposed, either for classes of theories or (by using general arguments) universally for all theories. Few are exact and inviolable. I present a new set of methods and sufficient conditions for deriving exact, rigorous, and sharp bounds on all coefficients of hydrodynamic dispersion relations, including diffusivity and the speed of sound. These general techniques combine analytic properties of hydrodynamics and the theory of univalent (complex holomorphic and injective) functions. Particular attention is devoted to bounds relating transport to quantum chaos, which can be established through pole-skipping in theories with holographic duals. Examples of such bounds are shown along with holographic theories that can demonstrate the validity of the conditions involved. I also discuss potential applications of univalence methods to bounds without relation to chaos, such as for example the conformal bound on the speed of sound.