Remote Nordic HET seminar: Sašo Grozdanov

Speaker: Sašo Grozdanov

Title: The complex structure of hydrodynamics: convergence, quantum chaos and bounds

Abstract: I will present three pieces of recent new insight into classical hydrodynamics, particularly into the complex analytic structure of hydrodynamic dispersion relations that express the mode's frequency as a function of its momentum. Crucially, one must treat both the frequency and the momentum as complex variables. Then, firstly, by recasting the problem of determining the dispersion relations as finding solutions to equations of complex spectral curves with certain critical points (analogously to treatments in algebraic geometry), one can show that hydrodynamic dispersion relations are Puiseux series with a finite radius of convergence. The radius can be explicitly computed in theories with holographic duals. Secondly, by analysing a dispersion relation and its corresponding correlation function in some holographic theory, one can compute the microscopic Lyapunov exponent and the butterfly velocity of quantum chaos through the phenomenon of pole-skipping. Thirdly, by treating the dispersion relations as univalent (complex holomorphic and injective) functions, one can derive rigorous and sharp bounds on all coefficients of the infinite hydrodynamic (Puiseux) series, including diffusivity and the speed of sound. Simplest examples of such bounds in holographic theories use pole-skipping to bound hydrodynamic transport in terms of quantities characterising quantum chaos (the Lyapunov exponent and the butterfly velocity).