Momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields

We present a holographic perspective on momentum transport in strongly coupled, anisotropic non-Abelian plasmas in the presence of strong magnetic fields. We compute the anisotropic heavy quark drag forces and Langevin diffusion coefficients and also the anisotropic shear viscosities for two different holographic models, namely, a top-down deformation of strongly coupled $\mathcal{N}=4$ super-Yang-Mills theory triggered by an external Abelian magnetic field, and a bottom-up Einstein-Maxwell-dilaton (EMD) model which is able to provide a quantitative description of lattice QCD thermodynamics with ($2+1$) flavors at both zero and nonzero magnetic fields. We find that, in general, energy loss and momentum diffusion through strongly coupled anisotropic plasmas are enhanced by a magnetic field being larger in transverse directions than in the direction parallel to the magnetic field. Moreover, the anisotropic shear viscosity coefficient is smaller in the direction of the magnetic field than in the plane perpendicular to the field, which indicates that strongly coupled anisotropic plasmas become closer to the perfect fluid limit along the magnetic field. We also present, in the context of the EMD model, holographic predictions for the entropy density and the crossover critical temperature in a wider region of the ($T$, $B$) phase diagram that has not yet been covered by lattice simulations. Our results for the transport coefficients in the phenomenologically realistic magnetic EMD model could be readily used as inputs in numerical codes for magnetohydrodynamics.

Figure 1

Normalized entropy density for the magnetic brane setup as a function of the dimensionless ratio $T/\sqrt{\mathcal{B}}$. In the inset we show the corresponding behavior for large magnetic fields.

Figure 2

Anisotropic drag force $F_{\mathrm{drag}}^{(v\parallel \mathcal{B})}$ in the magnetic brane model normalized by the isotropic SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 3

Anisotropic drag force $F_{\mathrm{drag}}^{(v\perp \mathcal{B})}$ in the magnetic brane model normalized by the isotropic SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 4

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\parallel v)}^{(v\parallel \mathcal{B})}$ in the magnetic brane model normalized by the SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 5

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\perp v)}^{(v\parallel \mathcal{B})}$ in the magnetic brane model normalized by the SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 6

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\parallel v)}^{(v\perp \mathcal{B})}$ in the magnetic brane model normalized by the SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 7

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\perp v,\perp \mathcal{B})}^{(v\perp \mathcal{B})}$ in the magnetic brane model normalized by the SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 8

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\perp v,\parallel \mathcal{B})}^{(v\perp \mathcal{B})}$ in the magnetic brane model normalized by the SYM result at zero magnetic field. Left: 3D plot as function of $\mathcal{B}/T^2$ and $v$. Right: As a function of $\mathcal{B}/T^2$ for some fixed values of $v$.

Figure 9

Ratio between the parallel and perpendicular shear viscosities in the magnetic brane model as a function of the dimensionless ratio $\mathcal{B}/T^2$.

Figure 10

Numerical bound on the maximum value for the initial condition $\mathcal{B}$ given some initial condition ${\varphi }_0$: initial conditions below the curve give asymptotically ${\mathrm{AdS}}_5$ solutions.

Figure 11

Thermodynamics of the magnetic EMD model at $B=0$ matched to lattice data in order to fix the free parameters of the holographic model. Top left: Entropy density. Top right: Speed of sound squared. Middle left: Pressure. Middle right: Internal energy density. Bottom left: Trace anomaly. Bottom right: Magnetic susceptibility.

Figure 12

Thermodynamics of the magnetic EMD model at nonzero $B$ compared to lattice data. Left: Entropy density as function of temperature for some fixed values of the magnetic field. Right: Pressure difference (in the direction of the magnetic field), $\mathrm{\Delta }p(T,B)\equiv p(T,B)-p(T_{\mathrm{ref}}=125\mathrm{MeV},B)$, as a function of temperature for some fixed values of the magnetic field.

Figure 13

New holographic predictions from the magnetic EMD model for the entropy density and the crossover temperature in an extended region of the ($T$, $B$) phase diagram not yet covered on the lattice. Top: Entropy density. Bottom: Crossover temperature extracted from the inflection point of $s/T^3$ compared to the corresponding lattice data.

Figure 14

Anisotropic drag force $F_{\mathrm{drag}}^{(v\parallel B)}$ in the magnetic EMD model normalized by the isotropic SYM result at zero magnetic field. Top: Results for the heavy quark velocity $v=0.50$. Bottom: results in the ultrarelativistic limit $v=0.99$.

Figure 15

Anisotropic drag force $F_{\mathrm{drag}}^{(v\perp B)}$ in the magnetic EMD model normalized by the isotropic SYM result at zero magnetic field. Top: results for the heavy quark velocity $v=0.50$. Bottom: results in the ultrarelativistic limit $v=0.99$.

Figure 16

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\parallel v)}^{(v\parallel B)}$ in the magnetic EMD model normalized by the SYM result at zero magnetic field. Top: Results for the heavy quark velocity $v=0.50$. Bottom: Results in the ultrarelativistic limit $v=0.99$.

Figure 17

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\perp v)}^{(v\parallel B)}$ in the magnetic EMD model normalized by the SYM result at zero magnetic field. Top: Results for the heavy quark velocity $v=0.50$. Bottom: results in the ultrarelativistic limit $v=0.99$.

Figure 18

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\parallel v)}^{(v\perp B)}$ in the magnetic EMD model normalized by the SYM result at zero magnetic field. Top: Results for the heavy quark velocity $v=0.50$. Bottom: results in the ultrarelativistic limit $v=0.99$.

Figure 19

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\perp v,\perp B)}^{(v\perp B)}$ in the magnetic EMD model normalized by the SYM result at zero magnetic field. Top: Results for the heavy quark velocity $v=0.50$. Bottom: Results in the ultrarelativistic limit $v=0.99$.

Figure 20

Anisotropic Langevin diffusion coefficient ${\kappa }_{(\perp v,\parallel B)}^{(v\perp B)}$ in the magnetic EMD model normalized by the SYM result at zero magnetic field. Top: Results for the heavy quark velocity $v=0.50$. Bottom: Results in the ultrarelativistic limit $v=0.99$.

Figure 21

Ratio between the parallel and the perpendicular shear viscosities in the magnetic EMD model as a function of $T$ and $B$.

Figure 22

Schematic representation of the anisotropic Langevin diffusion coefficients computed in this work.