The complex life of hydrodynamic modes

Abstract

We study analytic properties of the dispersion relations in classical hydrodynamics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in ${\cal N}=4$ supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in conformal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations.

Figures (18)

• Figure 1: Dispersion relations of the hydrodynamic modes in the strongly coupled ${\cal N}=4$ SYM theory, obtained using the dual holographic description. The dispersion relations are plotted in terms of dimensionless $\mathfrak{w} \equiv\omega/2\pi T$ and $\mathfrak{q}\equiv |{\bf q}|/2\pi T$, with complex $\mathfrak{w}$ as functions of real positive $\mathfrak{q}$. The left panel shows $\mathfrak{w}_{\rm shear}(\mathfrak{q})$ for the shear mode, the right panel shows $\mathfrak{w}_{\rm sound}^{+}(\mathfrak{q})$ for one of the two sound modes. In the left panel, the actual $-{\rm Im}\,\mathfrak{w}_{\rm shear}(\mathfrak{q})$ for the shear mode is shown by the solid red curve, and the analytic hydrodynamic approximation to $O(\mathfrak{q}^8)$ (computed in sec. \ref{['shear-n=4-hydro']}) is shown by the dashed blue curve. The blue dot indicates the pole-skipping point at $\mathfrak{q}_* =\sqrt{3/2}$, $\mathfrak{w}_* =-i$, discussed in sec. \ref{['sec:Chaos']}. The right panel shows ${\rm Re}\,\mathfrak{w}_{\rm sound}^{+}(\mathfrak{q})$ (solid red curve) and $-{\rm Im}\,\mathfrak{w}_{\rm sound}^{+}(\mathfrak{q})$ (dashed red curve) for the "+" sound mode. The straight dotted line indicates the light cone $\hbox{Re}\,\mathfrak{w} =\mathfrak{q}$.
• Figure 2: The analytically continued sound mode frequencies in the strongly coupled ${\cal N}=4$ SYM theory, obtained using the dual holographic description. The dimensionless frequencies $\mathfrak{w}_{\rm sound}^{\pm}$ of the two sound modes are plotted for purely imaginary dimensionless spatial momentum $\mathfrak{q}$, with the "+" branch in red and the "$-$" branch in blue. The frequencies $\mathfrak{w}_{\rm sound}^{\pm}$ are purely imaginary at imaginary $\mathfrak{q}$. At small momenta, the curves are linear with slopes $\pm v_s$, with $v_s=1/\sqrt{3}$. The curves pass through pole-skipping points $(\mathfrak{q}_*,\mathfrak{w}_*) =(\pm i\sqrt{3/2}, i)$ indicated by the blue dots.
• Figure 3: The first two (closest to the origin) poles of the retarded function of $T^{xy}$ in the strongly coupled ${\cal N}=4$ SYM theory, obtained using the dual holographic description. The locations of the poles are plotted as functions of the dimensionless wave vector for $\mathfrak{q}$ purely imaginary, with ${\rm Re}\, \mathfrak{w}$ shown in blue, and ${\rm Im}\, \mathfrak{w}$ shown in red. The dots indicate the points $(\mathfrak{q}_*,\mathfrak{w}_*)=(\pm i\sqrt{3/2},-i)$, where the response function of $T^{xy}$ exhibits pole-skipping.
• Figure 4: The Newton polygon for the sound mode.
• Figure 5: The approximations to the exact position of the critical point $|\mathfrak{q}_{\rm c}^2|=1/4$ in the holographic model with broken translation symmetry determined from the hydrodynamic algebraic curves $F_k(\mathfrak{q}^2,\mathfrak{w})=0$ as a function of $k$.