Holography and hydrodynamics with weakly broken symmetries

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.

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 nonhydrodynamic 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 $⟨P⟩$ 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 (2.12) 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 (2.12) 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 $\mathfrak{D}/\tau =\eta /((ϵ+p)\tau )=1/2$. 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 $\mathfrak{D}/\tau =\eta /((ϵ+p)\tau )=1/2$. Note that $\mathrm{Im}[{\omega }_1]=\mathrm{Im}[{\omega }_2]$.

Figure 6

The matched asymptotic expansion relies on simplifying the bulk equations in two separate limits (near-horizon and near-boundary), and on the overlap of the regions where these two different expansions are valid. The overlap appears very close (but not too close) to the horizon.