Holography and hydrodynamics with weakly broken symmetries

TL;DR

<3-5 sentence high-level summary>This paper develops quasihydrodynamics, a systematic framework for hydrodynamics with weakly broken symmetries, and shows how long-lived, approximately conserved quantities extend the hydrodynamic description. It provides a holographic algorithm to derive linearized quasihydrodynamic equations by matching outer boundary, near-horizon inner, and intermediate regions, allowing controlled resummations of frequency-dependent responses. The authors apply the method to two holographic realizations: magnetohydrodynamics with dynamical photons and Müller-Israel-Stewart theory arising from higher-derivative gravity, demonstrating the existence of dynamical photons and MIS-like relaxation from first principles. These results unify a broad class of phenomenological theories under a single holographic quasihydrodynamic framework and offer concrete tools for exploring weakly broken-symmetry dynamics in strongly coupled systems.

Abstract

Hydrodynamics is a theory of long-range excitations controlled by equations of motion that encode the conservation of a set of currents (energy, momentum, charge, etc.) associated with explicitly realized global symmetries. If a system possesses additional weakly broken symmetries, the low-energy hydrodynamic degrees of freedom also couple to a few other "approximately conserved" quantities with parametrically long relaxation times. It is often useful to consider such approximately conserved operators and corresponding new massive modes within the low-energy effective theory, which we refer to as quasihydrodynamics. Examples of quasihydrodynamics are numerous, with the most transparent among them hydrodynamics with weakly broken translational symmetry. Here, we show how a number of other theories, normally not thought of in this context, can also be understood within a broader framework of quasihydrodynamics: in particular, the Müller-Israel-Stewart theory and magnetohydrodynamics coupled to dynamical electric fields. While historical formulations of quasihydrodynamic theories were typically highly phenomenological, here, we develop a holographic formalism to systematically derive such theories from a (microscopic) dual gravitational description. Beyond laying out a general holographic algorithm, we show how the Müller-Israel-Stewart theory can be understood from a dual higher-derivative gravity theory and magnetohydrodynamics from a dual theory with two-form bulk fields. In the latter example, this allows us to unambiguously demonstrate the existence of dynamical photons in the holographic description of magnetohydrodynamics.

Figures (6)

• Figure 1: A schematic depiction of the ranges of validity of hydrodynamics and quasihydrodynamics for two distinct spectra plotted on the complex frequency $\omega$ plane. In the left panel, relaxation times of all non-hydrodynamic modes are comparable and the IR part of the spectrum is dominated by hydrodynamic modes. There is no quasihydrodynamic regime as $\tau_1 \sim \tau_2$. In the right panel, $\tau_1$ is much larger than the other relaxation times ($\tau_1 \gg \tau_2$) and so the hydrodynamic regime (shaded pink) has a greatly reduced regime of validity. Because $\tau_1 \gg \tau_2$, if we include the resulting long-lived mode in our effective theory, we obtain an improved quasihydrodynamic theory which is valid in a parametrically larger regime (shaded yellow). In each of the plots, the black circle depicts the hydrodynamic mode with the usual diffusive decay rate $\mathfrak{D} k^2$. The red circle is the massive mode $\langle P \rangle$ with the longest relaxation time $\tau_1$; blue circles depict modes with faster relaxation times $\tau_2,\,\tau_3,$ etc.
• Figure 2: The collision of the diffusive and gapped modes from the dispersion relation \ref{['eq:omegafick2']} is plotted on the complex $\omega\tau$ plane for a choice of $\mathfrak{D}/\tau = 1/2$. Arrows indicate the direction of movement of the poles as $k\tau$ is increased. The poles start on the imaginary axis, collide, and move off the axis.
• Figure 3: Plots of the real and imaginary parts of the two dispersion relations in \ref{['eq:omegafick2']} for a choice of $\mathfrak{D}/\tau = 1/2$.
• Figure 4: The collision of the MIS sound channel dispersion relations plotted on the complex $\omega\tau$ plane for a choice of $\mathfrak{D}/\tau = \eta/((\varepsilon + p) \tau)= 1/2$. Arrows indicate the direction of movement of the poles as $k\tau$ is increased.
• Figure 5: Plots of the real and imaginary parts of the sound dispersion relations for a choice of $\mathfrak{D}/\tau = \eta/((\varepsilon + p) \tau)= 1/2$. Note that $\text{Im}[\omega_1] = \text{Im}[\omega_2]$.