Thermodynamics and Bulk Viscosity of Approximate Black Hole Duals to Finite Temperature Quantum Chromodynamics

We consider classes of translationally invariant black hole solutions whose equations of state closely resemble that of QCD at zero chemical potential. We use these backgrounds to compute the ratio $\zeta /s$ of bulk viscosity to entropy density. For a class of black holes that exhibits a first-order transition, we observe a sharp rise in $\zeta /s$ near $T_c$. For constructions that exhibit a smooth crossover, like QCD does, the rise in $\zeta /s$ is more modest. We conjecture that divergences in $\zeta /s$ for black hole horizons are related to extrema of the entropy density as a function of temperature.

Figure 1

Left: A comparison of different $c_s^2(T)$ curves. The solid red curve corresponds to the potential (9) (type I black holes). The dashed magenta curve corresponds to the potential (12) (type II black holes). The dotted-dashed orange curve corresponds to the potential in (21) (type III black holes) with $a=1$ and $b$ adjusted so that the dimension of the operator dual to $\varphi$ is $\Delta \approx 3.93$. We also show lattice results for pure glue (solid black curve), based on Refs. 17, 32, lattice results for $2+1$-flavor QCD (open black squares), where we set $T_c=187\mathrm{MeV}$, as estimated from the halfway point of the initial rise in an $(ϵ-3p)/T^4$ curve from Ref. 33, and results from a $2+1$-flavor quasiparticle model [12]. Right: A comparison of $\zeta /s$ results for the black holes described above (see legend) and the results of Refs. 23, 24. Lattice results for pure glue from Ref. 24 are shown in blue. The solid curves correspond to the sum rule result for QCD with $2+1$ flavors from Ref. 23, using three representative values of the frequency parameter ${\omega }_0$.