Linear response beyond hydrodynamic poles

TL;DR

The paper develops a linearised effective theory that faithfully reproduces the Mittag-Leffler expansion of a conserved U(1) current with an arbitrary finite set of simple poles, while preserving hydrostaticity and standard thermodynamics. By treating time derivatives non-perturbatively through poles and organizing spatial derivatives perturbatively, it promotes the spatial current to a dynamical variable governed by a finite product of first-order operators, enabling exact matching to the pole structure and a controlled holomorphic sector. The framework is then tested in a holographic D3-D5 probe-brane model at finite density, where quasihydrodynamics emerges as a truncation effect in the pole expansion; AC conductivities, susceptibilities, and quasinormal modes are shown to align with the holographic approximant. The work offers a precise, non-perturbative handle on extended hydrodynamics and outlines future directions for nonlinear extensions, higher-form reformulations, and driven steady states.

Abstract

We consider the problem of writing an effective, linearised theory in small derivatives that reproduces the Mittag-Leffler expansion of a charge current correlator with an arbitrary number of simple poles. We demonstrate how such a framework: can be compatible with hydrostaticity without modification of thermodynamics, properly accounts for the differing notions of smallness in time and space derivatives including setting the lowest order effective equation of motion, and corrects the effective equations in derivatives. As an application, we apply the results to charge fluctuations of the D3/D5 probe brane and quantify how the transport coefficients behave when quasihydrodynamics emerges at large charge density.

Figures (7)

• Figure 1: A schematic diagram showing the distribution of poles (blue crosses) in the longitudinal conductivity at some charge density. The labels indicate the diffusive pole $\omega_{\mathfrak{D}}= - i \mathfrak{D}(\vec{k}^2) \vec{k}^2$ and the first gapped pole $\omega = - i \Gamma(\vec{k}^2)$. The dashed circle indicates the disc of convergence for the series expansion in frequency of the holomorphic part of the usual hydrodynamic charge conductivity \ref{['Eq:Infiniteseries']}.
• Figure 2: Difference of the position of the gapped pole obtained from the holographic approximant and the absolute value of the imaginary non-hydrodynamic quasi-normal mode lying closest to the origin, obtained from a shooting method. The difference between the shooting method and the holographic approximant is tiny. We find excellent agreement on the order of $\sim 10^{-56}$, where the two outliers that are not captured in the plot at ln$(\tilde{\rho}) = 2,3$ are of order $\sim 10^{-58}$ and $\sim 10^{-56}$, respectively, and thus lie outside the plotting range.
• Figure 3: Flow of the radius of convergence of the Taylor expansion of the holographic approximant around $\omega=0$ (equivalent to the gapped pole position) plotted against ln$(\tilde{\rho})$. (a) Shows the full range from $e^{-14}$ to $e^{14}$, while (b) shows a close-up for high $\tilde{\rho}$ values from $e^{8}$ to $e^{14}$.
• Figure 4: First coefficients of the small $\vec{k}$ expansion of (a) the charge susceptibility and (b) the magnetic susceptibility against the charge density. We find that the coefficients of the charge susceptibility grow like $\sim \tilde{\rho}^{1/2-n}$ for large charge density, while the magnetisation susceptibilities grow like $\sim \tilde{\rho}^{-1/2-n}$.
• Figure 5: Critical radius of the wavevector against the charge density for a small $\tilde{k}$ expansion of the charge susceptibility $\chi_{\mathrm{EE}}^{(\mathrm{L})}(\tilde{k}^2)$ and the magnetisation susceptibility $\chi_{\mathrm{BB}}(\tilde{k}^2)$. The radii grow as $\sqrt{\tilde{\rho}}$. The green line shows the critical radius for the zero sound mode.