Thermoelectric conductivities, shear viscosity, and stability in an anisotropic linear axion model

We study thermoelectric conductivities and shear viscosities in a holographically anisotropic model, which is dual to a spatially anisotropic $\mathcal{N}=4$ super-Yang-Mills theory at finite chemical potential. Momentum relaxation is realized through perturbing the linear axion field. Ac conductivity exhibits a coherent/incoherent metal transition. Deviations from the Wiedemann-Franz law are also observed in our model. The longitudinal shear viscosity for prolate anisotropy violates the bound conjectured by Kovtun-Son-Starinets. We also find that thermodynamic and dynamical instabilities are not always equivalent by examining the Gubser-Mitra conjecture.

Figure 1

The metric functions for $a=64.06$, $Q=9.76$ (left), which corresponds to the prolate anisotropy; and $a=1.2i$, $Q=1/10$ (right), with $u_H=1$, which depicts the oblate anisotropy.

Figure 2

Left: Dc conductivity as a function of the temperature corresponding to the smaller-horizon-radius branch, where we set $q=1.414$ and $a=0.1$. Middle: Dc conductivity as a function of the temperature in the thermodynamically stable regime, where we set $q=1.414$ and $a=0.1$. Right: The resistivity $\rho \sim {\rho }_{\mathrm{MIR}}T$ shows its linear temperature dependence in the small-horizon regime and that it can violate the Mott-Ioffe-Regel limit at high temperatures.

Figure 3

The optical conductivity as a function of frequency for different wave numbers $a/\mu =0.1$, 0.2, 0.5, 0.7, 0.8, 1.0 from left to right. The temperature is fixed at $T/\mu =0.2658$. The dots are data, and the solid lines are fit to the Drude formula.