Critical behaviour of hydrodynamic series

TL;DR

This paper analyzes the critical behavior of hydrodynamic modes in a strongly coupled gauge theory near a phase transition, computing transport coefficients and convergence properties up to third order in gradient expansion.

Contribution

It provides the first detailed calculation of higher-order hydrodynamic transport coefficients and mode convergence near a critical point in a top-down holographic model.

Findings

All hydrodynamic quantities share the same critical exponent near the transition.

The radius of convergence of hydrodynamic modes is determined for spin-1 and spin-2 sectors.

A relation between symmetry enhancement and vanishing third-order transport coefficient is established.

Abstract

We investigate the time-dependent perturbations of strongly coupled $\mathcal{N} = 4$ SYM theory at finite temperature and finite chemical potential with a second order phase transition. This theory is modelled by a top-down Einstein-Maxwell-dilaton description which is a consistent truncation of the dimensional reduction of type IIB string theory on AdS$_5\times$S$^5$. We focus on spin-1 and spin-2 sectors of perturbations and compute the linearized hydrodynamic transport coefficients up to the third order in gradient expansion. We also determine the radius of convergence of the hydrodynamic mode in spin-1 sector and the lowest non-hydrodynamic modes in spin-2 sector. Analytically, we find that all the hydrodynamic quantities have the same critical exponent near the critical point $\theta = 1/2$. Moreover, we establish a relation between symmetry enhancement of the underlying theory and vanishing the only third order hydrodynamic transport coefficient $\theta_1$, which appears in the shear dispersion relation of a conformal theory on a flat background.