Short-lived modes from hydrodynamic dispersion relations

TL;DR

The paper investigates the shear-diffusion mode in holographic relativistic hydrodynamics by generating a high-order $q$-series for the dispersion relation and showing that summing across branch cuts extends the series onto multiple sheets. The authors identify branch points at $q=\pm i q_\ast$ that bound the hydrodynamic series and demonstrate, using variable transformations and Padé resummation, that the hydrodynamic data can reproduce frequencies of non-hydrodynamic quasinormal modes on higher sheets. This approach indicates that the complete spectrum of gravitational quasinormal modes may be encoded in the hydrodynamic derivative expansion. Such a method offers a pathway to extract excited-state information from ground-state hydrodynamics, with potential applicability to a wide range of holographic and CFT systems.

Abstract

We consider the dispersion relation of the shear-diffusion mode in relativistic hydrodynamics, which we generate to high order as a series in spatial momentum q for a holographic model. We demonstrate that the hydrodynamic series can be summed in a way that extends through branch cuts present in the complex q plane, resulting in the accurate description of multiple sheets. Each additional sheet corresponds to the dispersion relation of a different non-hydrodynamic mode. As an example we extract the frequencies of a pair of oscillatory non-hydrodynamic black hole quasinormal modes from the hydrodynamic series. The analytic structure of this model points to the possibility that the complete spectrum of gravitational quasinormal modes may be accessible from the hydrodynamic derivative expansion.

Figures (6)

• Figure 1: The sequence $1+r_n$, for $r_n$ defined in \ref{['rn']} as ratios of successive terms in the hydrodynamic expansion. On this log-log plot $1+r_n$ is shown to converge linearly to $0$, indicated by the red line whose slope is $-1$. This confirms that the convergence of the hydrodynamic expansion is set by the scale $q_\ast$ defined in \ref{['qast']}.
• Figure 2: Poles and zeros of the Padé approximant, ${\cal P}_q$, to the hydrodynamic Taylor expansion of the shear-diffusion mode dispersion relation $\omega(q)$ in holography. The alternating sequence of poles and zeros along a radial ray is indicative of a branch point at $q = i q_\ast$, the closest non-analytic feature to the hydrodynamic limit $\omega = q=0$.
• Figure 3: Extending the hydrodynamic derivative expansion to a second sheet. The black points in all panels show the exact values, obtained numerically via a standard shooting calculation, and without invoking a derivative expansion. The coloured curves are as follows: Panel a) shows the hydrodynamic Taylor expansion, which converges only up to the branch point shown. Panel b) shows the hydrodynamic expansion summed as a Padé approximant ${\cal P}_z$ of a complex variable $z$ as defined in \ref{['zdef']}, producing two sheets (shown in lime and magenta respectively) which show excellent agreement up to a second branch point. Panels c) & d) show the Padé approximant ${\cal P}_u$ of a complex variable $u$ as defined in \ref{['udef']}, extending past another branch point and allowing the summed hydrodynamic expansion to describe the real line. The error in the empirically inferred second branch point of panel c) is damped as the real line is approached.
• Figure 4: Poles and zeros of the Padé approximant, ${\cal P}_z$, which is constructed using the variable $z$ (defined in \ref{['zdef']}) which covers more than one sheet. The outer dotted line corresponds to the original branch point discussed in section \ref{['convergence']}, whilst the inner dashed line corresponds to another branch point we must conquer to reach the real line, as indicated by the sequence of poles and zeros extending inwards from it, corresponding to the features in the magenta line, panel b) of figure \ref{['quad']}.
• Figure 5: Im$\omega(q)$ for two sheets constructed from a summation of the hydrodynamic series for the dispersion relation of the shear-diffusion mode in holography. The upper red dot corresponds to the hydrodynamic point, $q=0$ where $\omega(0) = 0$ on that sheet. The lower red dot corresponds to a non-hydrodynamic mode at $q=0$, where $\omega(0)$ on that sheet corresponds to the frequency of a non-hydrodynamic mode, given in table \ref{['nonhydrotable']}. The solid blue and black lines correspond to branch cuts. A comparison with the exact result on the Im$q$ axis is given in figure \ref{['quad']}.